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Bjiffii 

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IN  MEMORIAM 
FLORIAN  CAJORl 


(J^Jl^i  (t^^      ^/ 


A 

TEXT  BOOK 


OF 


Elementary  Mechanics. 


FOR  THE  USE  OF 


COLLEGES    AND    SCHOOLS. 


EDWARD  S.  DANA, 

C^Professor  ofJPhysks  in  Yale  College^) 
TWELFTH    EDITION. 


SEVENTH    THOUSAND 


NEW  YORK: 

JOHN   WILEY  AND  SONS, 

53  East  Tenth  Street 

1894. 


COPTBIGHT,  1881, 

Bt  lajWAED  S.  DANA. 


CAJORI 


PEEFAOE. 


The  writer  has  been  induced  to  prepare  this  Text 
Book  because  of  his  inability  to  find,  among  the  many 
excellent  works  on  Mechanics,  one  that  was  thoroughly 
adapted  to  his  special  wants,  and  because  it  has  seemed 
to  him  probable  that  similar  needs  must  have  been  felt 
by  other  instructors. 

The  chief  aim  has  been  to  present  the  fundamental 
principles  of  the  subject  in  logical  order,  and  in  as  clear, 
simple,  and  concise  a  form  as  possible,  yet  without  any 
sacrifice  of  strict  accuracy.  For  the  sake  of  making 
the  portions  of  the  subject,  which  necessarily  involve 
some  difficulty,  more  intelligible  to  beginners,  and  also 
to  increase  the  interest  of  the  general  principles  demon- 
strated by  showing  something  of  their  practical  bearings, 
simple  illustrations  have  been  introduced  rather  more 
fully  than  usual;  these  are  sometimes  given  in  a  few 
words,  sometimes  in  more  extended  form.  This  has 
led  to  a  slight  expansion  of  the  book  in  size,  but  does  not 
proportionately  increase  the  time  required  to  master  it. 
The  general  scheme  is  not  more  extended  than  the  sub- 
ject demands,  and,  if  time  is  limited,  the  instructor 
can  readily  select  those  articles  whose  omission  will  not 
interfere  with  the  completeness  of  the  study.     [Some 


Ql 1 282 


IV  PREFACE. 

of  the  articles  which,  if  necessary,  may  be  omitted,  are: 
26,  28,  36,-43,  45,  50,  51,  70,  75,  76,  81,  100,  105,  106, 
129,  131  a,  h  c,  135,  139,  142,  146,  149,  155,  158,  171, 
177,  189,  190, 193, 196,  197,  198, 199,  206,  208,  211,  213, 
214,  215,  216,  224,  225,  230,  231,  235,  239,  241,  242, 
243,  250,  251.  To  this  list,  which  might  be  somewhat 
extended,  can  be  added  the  matter  in  fine  print.] 

The  study  of  Elementary  Mechanics  is  one  of  very 
great  value  in  a  course  of  liberal  education.  The  sub- 
ject, when  properly  presented,  affords  an  excellent  kind 
of  mathematical  training,  which  is  suitable  to  all  grades 
of  students ;  it  furnishes  applications  of  the  principles 
and  methods  previously  learned  in  Geometry  and  Trigo- 
nometry, and  further  has  the  advantage  of  dealing  with 
the  real  phenomena  of  nature,  and  is  not  confined  to 
abstract  principles;  it  is  also  a  necessary  introduction  to 
Physics.  With  reference  to  the  last  end,  the  chapter 
on  Work  and  Energy  has  been  expanded  to  considerable 
length,  but  not  greater  than  the  importance  of  those 
principles  justifies. 

The  book  is  limited  to  the  Mechanics  of  Solids,  be- 
cause that  forms  a  complete  subject  by  itself,  while  an 
equally  extended  discussion  of  Liquids  and  Gases  would 
be  obviously  out  of  the  question.  Moreover,  the  latter 
subjects  are  treated  at  length  from  their  experimental 
side  in  Text  Books  on  Physics,  and  that  is  about  all  that 
the  general  student  requires. 

Examples  are  given  at  the  end  of  each  division  of  th^. 
subject,  designed  to  show  the  most  important  api)lica 
tions  of  the  principles;  answers  to  them  Avill  be  found 
on  pages  279-286.  By  simply  changing  the  numbers  in 
these  examples  they  may  be  increased  indefinitely  with- 
out adding  seriously  to  the  work  of  the  instructor.    The 


PREFACE.  V 

examples  are,  in  general,  arranged  so  as  to  involve  but 
little  mechanical  labor  of  computation — long  calcula- 
tions tending  rather  to  obscure  than  make  clear  the 
principles  for  the  illustration  of  which  the  examples  are 
given.  As  it  is  desirable  that  every  student  should  have 
some  knowledge  of  the  common  metric  units,  a  series  of 
examples  introducing  them  is  added  at  the  close  of  the 
volume. 

The  author  takes  pleasure  in  acknowledging  his  in- 
debtedness to  Prof.  H.  A.  Newton,  who  has  rendered 
him  very  important  aid  while  the  book  was  passing 
through  the  press,  and  also  to  Dr.  J.  J.  Skinner,  of  the 
Sheffield  Scientific  School,  for  numerous  valuable  sug- 
gestions. 

New  Haven,  Conn.,  January  1,  1881. 


TABLE   OF  OOl^TENTS. 


INTRODUCTION. 


1.  Matter.  2.  Body;  particle.  3.  Molecule.  4.  Physical  science. 
5.  Atom.  6.  Chemistry.  7.  States  of  matter:  solid,  liquid, 
gaseous.  8,  9.  Properties  of  matter.  10.  Mechanics:  Kine- 
matics, Dynamics  (or  Kinetics),  Statics Pages  1-5 

Brief  explanation  of  the  metric  system Pages  5,  6 


CHAPTER  I.— KINEMATICS. 

Motion  and  Best — Kinds  of  Motion. 

11.  Motion  and  rest.  12.  Kinds  of  motion.  13.  Motion  of  trans- 
lation. 14.  Motion  of  rotation.  15.  Translation  and  rotation. 
16.  Path  of  a  particle  or  body Pages  7-9 

Uniform  Motion. 

17.  Uniform  motion;  constant  velocity.     18.  Constant  angular 
velocity.     19.  Space  passed  over  in  uniform  motion.     20.  Geo- 
metrical representation  of  velocity.     21.  Geometrical  represen- 
tation of  the  space  passed  over  in  uniform  motion.  .Pages  9-12 
I.   Examples  :    Uniform    motion    of    translation    or    rota- 
tion   Pages  12,  13 

Varied  Motion — Acceleration. 

22.  Varied  motion;  variable  velocity.  23.  Acceleration.  24. 
Velocity  acquired  in  uniformly  accelerated  motion.  25.  Space 
passed  over  in  uniformly  accelerated  motion.  26.  Average 
velocity.    27,  28.  Formulas  for  accelerated  motion.  .Pages  13-20 


Vm  CONTENTS. 

II.  Examples  :  Uniformly  accelerated  motion  ;  Falling 
bodies,  etc Pages  20-23 

Composition  and  Resolution  of  Velocities —  Uniform  Motion. 

29.  Composition  of  motions  in  general.  30.  Resultant  and  com- 
ponent velocities.  31.  Composition  of  constant  velocities  in 
the  same  straight  line.  32.  Composition  of  two  constant 
velocities  not  in  the  same  straight  line.  33.  Parallelogram 
OF  Velocities.  34,  35.  Calculation  of  the  magnitude  and 
direction  of  the  resultant  velocity.  36.  Illustrations.  37. 
Composition  of  several  constant  velocities.    38.  Resolution  of 

velocities Pages  22-29 

III.,  IV.  Examples:  Composition  and  resolution  of  con- 
stant velocities Pages  30-32 

Composition  and  Resolution  of  Accelerations. 

39.   Composition  and  resolution  of  accelerations.      40.  Motion 

down  an  inclined  plane Pages  32,  33 

V.  Examples:  Bodies  falling  down  an  inclined  plane. 

Pages  33,  34 

Composition  of  Uniform  and  Accelerated  Motion  in  the  same 
Straight  Line. 

41,  42.  Composition  of  uniform  and  accelerated  motion  in  the 
same  straight  line.  43.  Geometrical  representation.  44. 
Motion  of  a  body  projected  vertically  upward.    45.  Projected 

up  or  down  an  inclined  plane Pages  3^39 

VI.,  VII.,  VIII.,  IX.  Examples:  Bodies  projected  vertically 
downward;  bodies  projected  vertically  upward;  bodies 
projected  up  or  down  a  smooth  inclined  plane;  bodies 
projected  against  friction Pages  39-42 

Composition  of  Uniform  and  Accelerated  Motion  not  in  the  same 
Straight  Line. 

46.  Composition  of  uniform  and  accelerated  motion.     47,  48,  49, 

50,  51.  Projectile Pages  42-50 

X.  Examples:  Projectiles Page  50 


CONTENTS.  IX 

CHAPTER  II.— DYNAMICS,  OR  KINETICS. 

Mass — Density —  Volume — Momentum. 

52.  Dynamics,  or  Kinetics.  53.  Mass  or  quantity  of  matter. 
54.  Mass  determined  by  weight.  55.  Distinction  between 
mass  and  weight.  56.  Relation  between  mass,  density,  and 
volume.     57.  Momentum Pages  51-55 

XI.  Examples:  Mass;  density;  volume ' Page  55 

58.  Definition  of   Force.     59.  Continued  and  impulsive  forces. 

60.  Effects  of  force  upon  a  free  body.     61.  Equilibrium.    63. 
Examples  of  force.    63,  64,  65.  Force  of  gravity.  ..Pages  55-62 

XII.  Examples:  Force  of  gravity Page  63 

JSlewton's  Laws  of  Motion. 

66.  Laws  of  Motion.  67.  First  law  of  motion  explained.  68. 
Second  law  of  motion  explained;  deduction  of  dynamical 
formulas.  69.  Third  law  of  motion  explained.  70.  Collision 
of  inelastic  bodies  Pages  63-71 

XIII.  Examples:  Collision  of  inelastic  bodies Page  72 

Measurement  of  Force. 
71.    Absolute    method    of    measuring    force.      72.    Gravitation 
method  of  measuring  force , Pages  72-75 

Problems  in  Dynamics. 

73.  Dynamical  formulas.  74.  Attwood's  machine.  75,  76.  Dy- 
namical problems Pages  75-80 

XIV.  Examples:  General  dynamical  problems.. Pages  80-83 


CHAPTER  III.— DYNAMICS— CENTRAL  FORCES. 

77.  Uniform  circular  motion ;  centrifugal  and  centripetal  forces. 
78.  Calculation  of  the  acceleration  of  a  central  force.  79. 
Illustrations  of  centrifugal  force.      80.  Centrifugal  force  due 

to  the  earth's  rotation Pages  83-88 

XV.  Examples:  Centripetal  and  centrifugal  forces. 

Pages  88,  89 


CONTENTS. 


CHAPTER  IV.— DYNAMICS— FRICTION. 

J.  Definition  of  friction.  83,  Reaction  of  smooth  surfaces. 
84.  Lubricators,  etc.  85.  Effects  of  friction.  86.  Kinds  of 
friction :  sliding  and  rolling.  87.  Fluid  friction.  88.  Laws  of 
friction.  89,  90.  Explanation  of  laws  of  friction.  91.  Co- 
eflBcient  of  friction.  92.  Angle  of  friction.  93.  Determination 
of  the  coefficient  of  friction.     94.  Examples  of  the , coefficient 

of  friction Pages  90-99 

XVI.  Examples:  Friction Pages  99,  100 


CHAPTER  v.— DYNAMICS-WORK  AND  ENERGY. 
A.  Mechanical  Work— Measurement  of  Work. 

95.  Definition  of  work.  96.  Examples  of  work.  97,  98.  Meas- 
urement of  work.     99.  Rate  of  work.      100.  Application  of 

principles  of  work Pages  101-105 

XVIL  Examples:  Work Pages  105, 106 

B.  Energy- Conservation  and  Correlation  op  Energy. 

101.  Definition  of  energy.  102.  Forms  of  energy.  103.  Kinetic 
and  potential  energy.  104,  105.  Measurement  of  energy. 
106.  Relation  of  kinetic  energy  to  momentum.  107.  Trans- 
formation of  kinetic  and  potential  energy.  108.  Apparent  loss 
of  visible  energy.  109.  Nature  of  heat.  110.  Examples  of 
the  production  of  heat  from  mechanical  energy.  111.  Definite 
relation  between  heat  and  mechanical  work.  112.  Conversion 
of  heat  into  w^ork.  113.  Other  forms  of  molecular  energy. 
114.  Examples  of  the  transformation  of  energy.  115.  Conser- 
vation of  energy.  116.  Terrestrial  stores  of  energy.  117. 
The  sun  as  the  ultimate  source  of  terrestrial  energy.      118. 

Dissipation  of  energy Pages  106-124 

XVIII.  Examples:  Potential  and  kinetic  energy. 

Pages  124-126 


CONTENTS.  XI 

CHAPTER  VI. -STATICS. 

IntrodiLctory. 

119.  Statics,  120.  Geometrical  representation  of  a  forcw.  121. 
Line  of  action  of  a  force.  122.  Transmission  of  a  force  in  its 
line  of  action.    123.  Body;  particle Pages  127,  128 

Composition  of  Forces  meeting  in  a  Point. 

124.  Composition  of  forces;  resultant  and  component  forces. 
125.  General  condition  of  equilibrium.  126.  Composition  of 
forces  having  the  same  line  of  action.  127.  Condition  of 
equilibrium  for  forces  having  the  same  line  of  action.  128. 
Parallelogram  of  Forces.  129.  Experimental  verification 
of  the  parallelogram  of  forces.  130,  131.  Calculation  of  the 
resultant.  132,  133.  Conditions  of  equilibrium  for  three  forces 
acting  on  a  particle.  134.  Composition  of  more  than  two  forces 
acting  in  the  same  plane  upon  a  particle.  135.  Forces  not  in 
the  same  plane.    136.  Condition  of  equilibrium  for  more  than 

three  forces  acting  on  a  particle Pages  129-142 

XIX.  Examples:  Parallelogram  of  forces Pages  142, 143 

Resolution  of  Forces. 

137.  Resolution  of  forces  in  general.  138.  Rectangular  com- 
ponents. 139.  Illustration  of  the  resolution  of  forces.  140. 
Resolution  of  forces  along  two  axes  at  right  angles  to  each 
other.  141.  Condition  of  equilibrium  for  three  or  more  forces 
acting  on  a  particle.     142.  Resolution  of  forces  along  three 

axes Pages  143-150 

XX.,  XXI,  Examples:  Resolution  of  forces.  .Pages  150-152 

Composition  and  Resolution  of  Parallel  Forces. 

143.  Parallel  forces.  144.  Like  parallel  forces.  145.  Unlike 
parallel  forces.  146.  Experimental  verification.  147.  Three 
or  more  parallel  forces.  148.  Three  parallel  forces  in  equi- 
librium.    149.  Resolution  of  parallel  forces.    150.  Couples. 

Pages  152-159 
XXII.  Examples:  Parallel  forces Pages  159, 160 


Xll  CONTENTS. 

Forces  tending  to  produce  Botation — Moments. 
151,  152.  Moment.     153.  Positive  and  negative  moments.     154 
Geometrical  representation  of  the  moment  of  a  force.     155. 
Proposition  in  regard  to  moments.     156.  Equality  of  moments; 
principle  of  the  lever.    157.  Free  and  constrained  body. 

Pages  161-167 

XXIII.  Examples:  Moments Page  167 

Summary  of  Conditions  of  Equilihrium. 

158.  Summary  of  conditions  of  equilibrium  for  forces  acting  on 
a  body  in  one  plane Pages  167, 168 

CHAPTER  VII.— STATICS— CENTRE  OF  GRAVITY. 
A.    Centre   op   Gravity  of    Bodies  —  Plane   and    Solid. 

159.  Definition  of  the  centre  of  gravity.  160.  Centre  of  gravity 
of  two  bodies.  161.  Centre  of  gravity  of  any  number  of 
bodies.  162.  Centre  of  gravity  of  a  straight  line.  163.  Centre 
of  gravity  of  a  plane  figure  determined  by  its  symmetry.  164. 
Centre  of  gravity  of  regular  polygons.  165.  Centre  of  gravity 
of  a  parallelogram.  166.  Centre  of  gravity  of  a  triangle.  167. 
Centre  of  gravity  of  a  solid  figure.  168.  Centre  of  gravity  of 
a  triangular  pyramid.  169.  Centre  of  gravity  of  any  pyramid. 
170.  Centre  of  gravity  of  a  cone.    171.  Problems.  .Pages  169-180 

XXIV.  Examples:  Centre  of  gravity Pages  180-182 

B.  Application  op  the  Principles  op  the  Centre  op 
Gravity— Equilibrium  and  Stability. 
172.  Conditions  of  equilibrium.  173.  Experimental  determina- 
tion of  the  centre  of  gravity.  174,  175.  Stable,  unstable,  and 
neutral  equilibrium.  176.  Stability  of  a  body  resting  on  a 
base.  177.  Conditions  upon  which  the  stability  of  a  body 
depends Pages  182-1^9 

XXV.  Examples:  Stability Pages  189, 190 

CHAPTER  VIII.— STATICS— MACHINES, 
178,  179,  180,  181.  Principle  of  work  as  applied  to  the  machines. 
182.   Virtual  velocities.      183.    The  machines  with  friction. 
184.  Simple  machines Pages  191-195 


CONTENTS.  Xlll 

I.  Lever. 

A.  General  Principles  of  the  Lever. 

185.  The  lever  defined.  186.  Relation  of  the  power  and  weight 
in  the  lever.  187.  Three  kinds  of  lever.  188,  189.  Illustration 
of  the  lever.     190.  The  lever  on  the  principle  of  work 

Pages  195-201 

B.  Some  Special  Applications  of  the  Principle  of  the  Lever. 

I.    BALANCE. 

191.  Balance.  193,  193.  Conditions  to  be  fulfilled  by  a  good 
balance Pages  202-305 

n.   STEELYARD. 

194.  Common  steelyard.  195.  Application  of  the  steelyard. 
196.  Danish  steelyard.     197.  Roberval's  balance.. Pages  305-310 

III.    TOGGLE-JOINT. 

198.  Relation  of  power  to  weight  in  the  toggle-joint.  199.  Stone- 
crushing  machine Pages  310-313 

IV.    COMPOUND  LEVERS. 

300.  Relation  of  power  to  weight  in  the  compound  levers.     301. 

Application  of  compound  levers Pages  313,  313 

XXVI.,   XXVII.,   XXVIII.    Examples:   Lever;    balance; 
steelyard Pages  313-315 

II.  Wheel  and  Axle. 

303.  Wheel  and  axle.  303,  304,  305.  Relation  of  power  to  weight 
in  wheel  and  axle.  306.  Wheel  and  axle  on  the  principle  of 
work.     307.  Applications  of  the  wheel  and  axle.     308.  Chinese 

windlass Pages  315-319 

XXIX.  Examples:  Wheel  and  axle Page  330 

III.  Toothed  Wheels. 

309.  Toothed  wheels.  310.  Relation  of  power  to  weight  with 
the  toothed  wheels.  311.  Toothed  wheels  on  the  principle  of 
work.  313,  313,  314,  315.  Applications  of  the  toothed  wheels. 
216.  Use  of  belts Pages  220-336 


XIV  CONTENTS. 

lY.  Pullet. 
217.  Pulley.  218.  Single  fixed  pulley.  219.  Single  movable 
pulley  with  parallel  strings.  220.  Single  movable  pulley  with 
inclined  strings.  221,  222,  223,  224.  Combinations  of  pulleys. 
225.  The  pulley  on  the  principle  of  work.  226.  Application 
of  the  pulley Pages  226-234 

XXX.  Examples:  Pulley Pages  234,  235 

V.  Inclined  Plane. 

227.  Inclined  plane.     228,  229,  230.  Relation  of  the  power  and 

weight  for  the  inclined  plane.     231.  The  inclined    plane  on 

the  principle  of  work.      232.    Applications  of    the    inclined 

plane Pages  235-241 

XXXI.  Examples:  Inclined  plane Pages  242,  243 

VI.  Wedge. 

233.  Wedge.  234.  Relation  of  the  power  to  the  weight  for  the 
wedge.  235.  The  wedge  on  the  principle  of  work.  236.  Appli- 
cation of  the  wedge Pages  243-245 

XXXII.  Examples:  Wedge Page  245 

VII.  Screw. 

237.  Screw,  238.  Relation  of  the  power  to  the  weight  for  the 
screw.  239.  The  screw  on  the  principle  of  work.  240.  Appli- 
cation of  the  screw.  241.  Micrometer  screw.  242.  Differen- 
tial screw.     243.  Endless  screw Pages  246-251 

XXXIII.  Examples:  Screw Page  251 

CHAPTER  IX.— PENDULUM. 

244.  Motion  in  a  vertical  circle.  245.  Motion  of  a  simple  pendu- 
lum. 246,  247.  Compound  pendulum.  248.  Application  of 
the  pendulum.  249,  250.  Values  of  Z  and  </.  251.  Other  appli- 
cations of  the  pendulum  Pages  252-261 

XXXIV.  Examples:  Pendulum Page  261 

Additional  Examples,  introducing  the  Metric  Units. 

Pages  263-278 
Answers  to  Examples Pages  279-291 


ELEMENTAEY  MECHANICS. 


INTRODUCTION". 


1.  Matter.  Matter  is  the  substance  of  whicli  bodies 
are  composed;  it  is  that  which  may  be  apprehended  by 
the  senses,  and  which  may  be  acted  upon  by  force. 

2.  Body-Particle.  A  tody  is  any  portion  of  matter 
which  is  bounded  in  every  direction.  A  material  parti- 
cle is  a  body  of  dimensions  so  small  that  it  is  unnecessary 
to  consider  the  differences  in  position  or  motion  of  its 
different  parts. 

In  many  cases  the  differences  in  the  relations  of  the 
parts  of  an  extended  body  are,  in  like  manner,  left  out 
of  account,  it  being  considered  as  a  single  unit,  and  then 
the  body  is  treated  as  a  particle. 

3.  Molecule.  The  smallest  portion  into  which  a 
given  kind  of  matter  can  be  conceived  to  be  divided, 
without  a  loss  of  its  properties,  is  called  the  molecule. 
The  molecule  is  an  ideal  unit,  the  existence  of  which  is 
believed  to  be  proved  by  experiment,  although  it  cannot 
be  by  direct  observation.  The  smallest  portion  of 
matter,  obtained  by  any  method  of  mechanical  subdivi- 
sion, would  consist  of  a  large  number  of  molecules. 

According  to  the  conclusions  of  Sir  William  Thomson, 
if  a  drop  of  water  were  to  be  magnified  to  the  size  of  the 
earth,  the  molecules,  of  which  it  is  made  up,  would  be 
coarser  than  fine  shot  and  probably  finer  than  cricket- 
balls. 


2  INTEODUCTION.  [4. 

4.  Physical  Science.  All  changes  which  involve  a 
material  body,  either  as  a  whole  or  with  respect  to  the 
relations  of  its  molecules,  are  considered  under  the  head 
of  Natural  Philosophy,  or  Physical  Science.  Thus,  the 
fall  of  a  body  to  the  earth;  the  flight  of  a  rifle-ball;  the 
ring  of  a  bell;  the  melting  of  iron,  and  its  contraction 
or  expansion  on  change  of  temperature,  its  magnetiza- 
tion— these  and  all  other  analogous  phenomena  are  in- 
cluded under  Physical  Science. 

6.  Atom.  Every  molecule  is  supposed  to  be  made  up 
of  one  or  more  indivisible  units  called  atoms  ( a  priv. 
and  ri/xvGD,  to  divide).  Thus,  the  smallest  conceivable 
particle,  or  molecule,  of  salt,  possessing  all  the  p'roper- 
ties  of  the  mass,  is  believed  to  consist  of  two  dissimilar 
atoms,  one  of  the  metal  sodium,  the  other  of  the  gas 
chlorine. 

6.  Chemistry.  All  phenomena  which  result  in  a  re- 
arrangement of  the  atoms  and  a  consequent  change  in 
the  molecules  of  a  body — that  is,  a  loss  of  identity  of  the 
substance  involved — belong  to  Chemistry.  For  example, 
the  change  of  ice  to  water,  or  of  water  to  steam,  involves 
no  change  in  the  molecules  but  only  in  their  mutual  re- 
lations and  position,  hence  these  phenomena  belong  to 
Physical  Science;  but  when  a  rearrangement  of  the  atoms 
takes  place  and  the  water  is  thus  decomposed  into  its 
constituent  gases,  hydrogen  and  oxygen,  this  lasfc  is  a 
chemical  change. 

The  molecule  is  the  physical  unit;  the  atom  is  the 
chemical  unit. 

7.  States  of  Matter.  Matter  may  exist  in  three  differ- 
ent  states  :  the  solid,  liquid,  and  gaseous  states. 

The  SOLID  is  characterized  by  a  greater  or  less  degree 


9.1  INTKODUCTION.  3 

of  rigidity.  The  molecules  are  bound  together  by  the 
molecular  force  of  attraction,  called  cohesion,  and  hence 
a  solid  body  tends  to  retain  its  own  shape. 

The  LIQUID  is  characterized  by  its  mobility;  the  mole- 
cules are  free  to  move  about  each  other,  and  the  liquid 
takes  the  shape  of  any  containing  vessel. 

The  GAS  is  characterized  by  its  tendency  to  indefinite 
expansion.  The  molecules  are  believed  to  be  in  rapid 
motion  and  constantly  coming  into  collision  and  then 
repelling  one  another,  so  that  a  gas  tends  to  occupy  a 
greater  volume,  and  hence  exerts  pressure  on  the  sides 
of  any  vessel  in  which  it  is  confined. 

The  term  fluid  is  sometimes  employed  to  include  both 
liquids  and  gases. 

Many  substances  may  under  varying  conditions  exist 
in  the  three  different  states  :  this  is  illustrated  by  the 
case  of  water,  which  is  a  solid — ice — below  the  freezing 
point,  a  liquid  at  ordinary  temperatures,  and  a  gas — steam 
— at  high  temperatures. 

8.  Properties  of  Matter.  All  forms  of  matter  possess 
the  essential  properties  of  extension,  impenetrability,  and 
inertia. 

(1)  Extension  :  Every  body  occupies  a  definite  por- 
tion of  space;  that  is,  it  has  length,  breadth,  and  thick- 
ness. 

(2)  iMPEiq-ETRABiLiTY  :  Two  forms  of  matter  cannot 
occupy  the  same  space  at  the  same  time. 

(3)  Inertia  :  Matter  has  no  power  to  change  its 
own  state  of  motion  or  rest,  hence  it  offers  an  apparent 
resistance  to  a  force  tendmg  to  change  its  state.  This  is 
further  explained  in  a  subsequent  article  (67). 

9.  Other  properties  of  matter  are — 

Porosity  :  The  molecules,  of  which  a  given  body  is 


4  INTRODUCTION",  [10. 

supposed  to  be  made  up,  are  believed  to  be  separated 
from  one  another  by  a  greater  or  less  space.  In  addi- 
tion to  these  true  or  physical  pores,  most  bodies  exhibit 
also  visible  open  spaces,  or  sensible  pores,  as  those  of  a 
sponge. 

Compressibility  :  A  body  may  be  made  by  pressure 
to  occupy  a  smaller  space  ;  this  is  a  direct  consequence 
of  its  porosity. 

Divisibility  :  A  given  kind  of  matter  admits  of  be- 
ing divided  into  a  very  great  number  of  parts. 

Elasticity  :  A  body,  whose  shape  has  been  altered  by 
a  force  acting  on  it,  tends  to  regain  its  shape  when  the 
force  ceases  to  act.  Solids  vary  widely  in  elasticity  :  for 
example,  compare  lead  and  steel,  or  clay  and  ivory. 
Liquids  and  gases  are  perfectly  elastic. 


10.  Mechanics  is  that  branch  of  Physical  Science 
which  considers  the  motion  and  equilibrium  of  bodies. 
Corresponding  to  the  three  states  of  matter,  the  subject 
of  Mechanics  is  divided  into 

(1)  Mechanics  of  solids. 

(2)  Mechanics  of  liquids,  including  Hydrostatics  and 
Hydrodynamics. 

(3)  Mechanics  of  gases,  or  Pneumatics. 

The  first  of  these  three  divisions,  which  forms  the 
subject  of  this  text-book,  is  further  divided  into  three 
parts,  Kinematics,  Dynamics  or  Kinetics,  and  Statics. 

Kinematics*  includes  the  discussion  of  abstract 
motion;  that  is,  of  the  motion  of  bodies  without  refer- 
ence to  their  mass  (quantity  of  matter),  or  to  the  force  or 

*  From  the  Greek  nivrjtxa,  motion. 


10.]  INTRODUCTIOTT.  O 

forces  which  cause  their  motion.  To  the  idea  of  space, 
involved  in  Geometry,  it  adds  that  of  time. 

Dynamics,*  or  Kinetics, f  embraces  the  discussion  of 
the  action  of  a  force,  or  of  forces,  in  producing  the  mo- 
tion of  bodies  of  known  mass. 

StaticsJ  discusses  the  action  of  forces  upon  bodies  in 
so  far  as  they  hold  the  body  acted  upon  at  rest;  that  is, 
in  equilibrium. 

*From  the  Greek  Svvajui<;,  power. 

f  From  xivEoo,  to  move, 

X  From  drarixo^  (idrrjixt),  causing  to  stand. 

METRIC  SYSTEM. 

UNITS  OF  LENGTH. 

ENGLISH  UNITS. 

Kilometer... 1000  meters.  3280.9  feet.            .62137   mile. 

Meter 39.37    inches.  3.281    feet. 

Decimeter 1  meter.  3.937      "          .3281     " 

Centimeter 01     '*  0.3937   "          .0328     " 

Millimeter  {mm)..  .001  "  0.0394   '♦          .00328   " 

UNITS  OF  VOLUME. 

DKY  MEASTJHE.      LIQUID  MEASUBE. 

Kiloliter  (or  Stere)  1  cub.  meter,       1.308  cub.  yds.    264.17  galls. 

Liter 1  cub.  decimeter,  0.908  quarts,     1.0567  quarts. 

Milliliter 1  cub.  centimeter,  0.061  cub.  in.    0.37  fl.  dr'hm. 

UNITS  OF  WEIGHT, 
VOLUME  OF  WATER  AVOIBDUPOIS    MEASURE. 

GIVING  THE  WEIGHT.  (1  lb.  =7000  grains.) 

Kilogram cub.  dec'm'r  or  liter.  15432.3  grains.    2.2046    lbs. 

Gram cub.  centimeter.  15.432  "        0.0022      " 

Milligram....  cub.  millimeter.  0.0154"       0.0000022" 


METEIC  SYSTEM. 

■^  Meter  =  1  Decimeter  =  10  Centimeters  =  100  Millimeters. 


Centimeters. 
10              9               S               f               6                5                4                3                2 
1                 1                 1                1                 1                 1        1         1        1         1        1 

1 

1 

lllllllll   lllllllll   lllllllll    llllllMI    lllllllll    lllllllll 

MIIIIIIMIMMIMIIMII 

JUI  )lll  U 

1     |l           1      1     1           1     j.l           1      1      1           1     1     1           1      1     1           1      1      1 

4                                        3                                          2                                          1 

Inches. 

1  1  • 

4  inches,  each  divided  into  eighths. 

The  Meier  is  the  length  at  0°  C  (temperature  of  melting  ice)  of  a 
certain  platinum  bar  kept  at  Paris.  It  was  intended  to  be  (and  is 
very  nearly)  equal  to  one  forty-millionth  part  of  the  earth's  cir- 
cumference about  the  poles.  It  is  the  only  arbitrary  unit  of  the 
Metric  System,  since  all  the  other  units  of  weight,  etc.,  are  directly 
deiived  from  it. 

Units  of  Length.  The  meter  is  divided  into  10  decimeters, 
into  100  centimeter>Sy  into  1000  millimeters.  Also,  10  meters=:l  deka- 
meier,  100  meters::;  1  hectometer^  1000  meters=l  kilometer.  (The 
same  prefixes  are  used  in  a  case  of  the  other  units,  with  a  similar 
signirication.) 

The  approximate  value  of  some  of  these  units  are  as  follows: — 
The  meter  is  a  little  longer  than  the  English  yard ;  it  is  very  nearly 
equal  to  3  feet  3f  inches.  The  millimeter  is  a  little  less  than  .04  of 
an  inch,  or  1  inch  is  a  little  more  than  25  millimeters.  (See  figure 
above.)    The  kilometer  is  about  f  of  a  mile. 

Unitb  of  Subface.  The  squares  of  the  units  of  length  are 
taken  as  the  units  of  surface.  The  principal  units  are  the  centare, 
or  squave  meter;  the  art  (  =  100  square  meters);  and  the  hectare 
(=10,000  square  meters);  the  hectare  is  equal  to  2.47  acres. 

Units  of  Volume.  The  cubes  of  the  units  of  length  are  taken 
as  the  units  of  volume  or  capacity.  The  principal  units  are  the 
cubic  meter  or  stere,  equal  to  1.3  cubic  yards;  the  cubic  decimeter 
or  liter,  which  is  a  little  larger  than  a  wine  quart;  and  the  cubic 
centimeter. 

Units  of  Weight.  The  weights  of  the  units  of  volume  of 
water  (at  4°C  =  39°. 2  F  when  it  has  its  greatest  density)  are  taken  as 
the  units  of  weight.  The  principal  units  of  weight  are  the  kilogram 
(or  kilo),  which  is  the  weight  of  a  liter  or  cubic  decimeter  of  water 
at  4°  C;  it  is  equal  to  2.2  pounds;  and  the  gram,  which  is  the 
weight  of  the  cubic  centimeter  of  water  at  4°  C;  it  is  equal  to  about 
15  grains. 

The  equivalents  of  the  important  metric  units  are  given  more 
exactly  on  the  preceding  page. 


CHAPTER  I.— KINEMATICS. 

Motion  and  Rest — Kinds  of  Motion, 

11.  Motion.  A  body  is  said  to  move  when,  in  succeS" 
BJve  intervals  of  time,  it  occupies  different  positions  with 
reference  to  some  other  body  considered  to  be  at  rest. 

The  terms  motion  and  rest  are  simply  relative,  for  the 
state  of  any  body  in  this  respect  can  be  judged  of  only 
by  comparing  it  with  some  other  body  or  bodies.  For 
example,  the  objects  on  the  deck  of  a  steamboat  may  be 
at  rest  with  reference  to  each  other  and  to  the  boat,  while 
they  are  in  motion  as  regards  the  neighboring  shore. 
Again,  two  trains  moving  side  by  side  at  the  same  speed 
may  seem  to  a  passenger  on  either  to  be  at  rest,  and  are 
actually  so  as  regards  each  other,  while  they  are  in  rapid 
motion  as  regards  the  ground  over  which  they  are  pass- 
ing. 

As  the  term  rest  is  ordinarily  employed  in  Mechanics, 
the  earth  is  used  as  the  basis  of  comparison,  and  in  this 
sense  bodies  are  said  to  be  at  rest  which  do  not  move 
with  reference  to  it,  as,  for  example,  the  buildings  in  a 
city.  It  is  to  be  remembered,  however,  that  the  earth 
itself  and  hence  all  objects  upon  it  are  really  moving 
very  rapidly  through  space.  In  fact  we  know  nothing 
of  absolute  rest,  for  all  bodies  of  which  we  have  any 
knowledge  are  in  motion. 

Further  than  this,  there  is  reason  to  believe  that,  in- 
dependent of  the  motion  of  the  bodies  themselves,  the 


8  KIlSrEMATICS.  [12. 

molecules  which  make  them  up  have  also  in  all  cases  a 
very  rapid  vibratory  motion  of  their  own.  Motion  ia 
then  the  actual  state  of  matter  so  far  as  we  know  it, 
while  the  rest  we  observe  is  only  apparent. 

12.  Kinds  of  Motion.  With  respect  to  its  direction,  a 
body  may  have  either  motion  of  translation  or  of  rota- 
tion. With  respect  to  its  rate,  the  motion  may  be  uni- 
form or  varied. 

13.  Motion  of  Translation.  If  the  motion  of  a  body 
is  such  that  every  point  in  it  has  the  same  velocity,  and 
every  straight  line  in  it  remains  parallel  to  itself,  the 
body  is  said  to  have  tnotion  of  translation.  This  is  illus- 
trated by  the  motion  of  a  sled  down  a  hill,  or  that  of  the 
body  of  a  carriage. 

The  motion  of  translation  of  a  particle  may  be  either 
(1)  rectilinear — that  is,  in  a  straight  line — or  (2)  curvi- 
linear, in  a  curved  line. 

14.  Motion  of  Rotation.  A  body  is  said  to  have  motion 
of  rotation,  or  simply  to  rotate,  when  it  moves  about  an 
axis  so  that  the  different  particles  describe  concentric 
circles  around  -it,  their  velocity  increasing  with  their  dis- 
tance from  the  axis.  Tliis  is  illustrated  by  the  turning 
of  a  wheel  on  its  axle,  or  the  spinning  of  a  top. 

16.  A  body  may  at  the  same  time  have  both  kinds  of 
motion.  For  example,  the  wheel  of  a  carriage  rotates 
about  its  axle,  and  also  moves  forward — that  is,  has 
motion  of  translation — with  the  rest  of  the  vehicle ;  if 
the  wheel  is  blocked,  as  in  descending  a  steep  hill,  then 
it  has  motion  of  translation  only.  Again,  the  earth  has 
a  motion  of  rotation  about  its  axis  and  also  of  transla- 
tion in  its  orbit  about  the  sun. 

In  the  statements  which  follow  in  regard  to  the  motion 


17.]  UNIFOEM   MOTION.  9 

of  bodies,  motion  of  translation  without  rotation  is 
always  to  be  understood  unless  it  is  distinctly  stated 
otherwise. 

As  explained  in  Art.  2,  the  term  body  may  be  used 
instead  of  particle,  when  the  body  is  considered  as  a 
unit,  any  distinction  between  the  position  or  motion 
of  the  different  parts  being  left  out  of  account ;  in  this 
sense  the  term  body  is  employed  in  the  following  arti- 


16.  Path  of  a  Particle  or  Body.  The  path  of  a  parti- 
cle, or  trajectory  as  it  is  sometimes  called,  is  the  con- 
tinuous line,  either  straight  or  curyed,  which  it  describes 
as  it  moves.  By  the  path  of  a  body  is  ordinarily  meant 
the  line  described  by  some  definite  point  in  it,  usually 
the  centre  of  gravity,  or  the  geometrical  centre. 


Uniform  Motion, 

17.  Xlniform  Motion.  The  motion  of  a  body  is  said  to 
be  uniform  if  it  moves  over  equal  spaces  in  equal  suc- 
cessive intervals  of  time,  however  small  these  be  taken. 
The  velocity,  or  rate  of  motion,  is  then  said  to  be  C07i' 
stant. 

Constant  velocity,  or  the  velocity  of  a  hody  moving 
uniformly,  is  measured  by  the  number  of  units  of  linear 
space  passed  over  in  the  unit  of  time. 

The  UNIT  OF  SPACE  commonly  employed  is  the  foot, 
and  of  time  the  second.  The  ujs'IT  of  velocity  is  then 
a  velocity  of  one  foot  per  second ;  this  is  a  compound 
unit  sometimes  called  a  foot-second.  Thus  a  constant 
velocity  of  10  would  mean  that  the  body  passed  over  10 
feet  in  each  successive  second. 


10  KINEMATICS.  [1 

Other    units  of   space  and  time  are  also  not  infr 

quently   employed  :    we  speak  of   the  yelocity  of  tl 

earth  in  its  orbit  as  19  miles  per  second  ;   of  a  train  i 

so  many  miles  an  hour,  and  so  on.    If  the  metric  systei 

is  made  use  of,  the  units  of  distance  belonging  to  it  mui 

be  taken ;  that  is,  the  millimeter,  meter,  kilometer,  et 

18.  Constant  Angular  Velocity.     The  statement  in  tl 

preceding  article  has  reference  only  to  linear  velocit; 

When,   however,   a  body  rotates  on  an  axis  and  eac 

particle  describes  a  circle  about  it,  it  is  often  convenie] 

to  have  an  expression  for  the  angular  velocity. 

The  angular  velocity  of  a  iod 

rotating  uniformly  about  an  axi 

is  measured  hy  the  angle  descrih 

in  the  unit  of  time  hy  a  radii 

moviyig  in  a  plane  perpeiidicuh 

to  the  axis  of  rotation.  Thisangl 

as  AGB  (Fig.  1),  is  expressed  n 

Fig.  1.  in  degrees  but  in  circular  measui 

(  AB 
that  is,  by  the  ratio  of  the  arc  to  the  radius  ( — —- y 

This  angular  velocity  is  usually  represented  by  the  lett 
OD.     Hence 

AB 
"^  =  -10' 

It  AC  =  r,  and  AB  =  v  =  the  linear  velocity,  then 

G3  =  — ,  and  .'.  V  =  Gor, 
r 

It  is  evident  that  the  angular  velocity  is  constant  f or  i 
parts  of  a  body  rotating  uniformly,  but  the  linear  ve^ 
city  increases  directly  with  the  distance  from  the  axis. 


81.1  UNIFORM   MOTION.  11 

19.  Space  passed  over  in  Uniform  Motion.  If  a  body 
moves  uniformly  for  a  time  i,  with  a  Telocity  v,  the 
space,  or  distance  {s),  passed  over  is  equal  to  the  product 
of  the  time  and  velocity  : 

s  =  vf; 

s  s 

also,  V  =  —,  and  t  =  —, 

t  V 

20.  Geometrical  Representation  of  Velocity.  The  ve-^ 
locity  of  a  body  may  be  represented  geometrically  by 
a  straight  line,  whose  direction  is  the  direction  of  the 
motion,  and  whose  length  is  taken  proportional  to  the 
velocity.     Thus  (Fig.  2),  if  the  motion  of  a  particle  be 


Fig.  2.  Fio.  8. 

in  the  direction  from  A  toward  0  with  a  velocity  of  20, 
and  in  another  independent  case  from  A  toward  B  with 
a  velocity  of  10,  then  these  lines,  if  proportional  to  20 
and  10  respectively,  may  be  taken  as  representing  these 
velocities  geometrically ;  here  obviously  AC  =  2AB. 

If  the  particle  moves  in  a  curved  path,  as  from  M 
toward  ^Y  (Fig.  3),  its  velocity  at  any  points,  as  A  and  G, 
will  be  represented  by  tangents  at  these  points,  AB  and 
CD,  whose  lengths  are  proportional  to  the  velocities 
respectively. 

21.  Geometrical  Representation  of  the  Space  passed 
over  in  Uniform  Motion.     The  space  passed  over  by  a 


12  KINEMATICS.  [21, 

body  moving  uniformly  for  a  given  time  may  be  repre- 
sented by  the  area  of  a  rectangle,  whose  adjacent  sides 
are  taken  proportional  respectively 
to  the  time  and  velocity,  each  in 
r  terms  of  its  own  unit.  Thus,  sup- 
pose a  body  to  move  for  t  seconds 
^  ^  ji      with  the  constant  velocity  v,   let 

Fro.  4.  AB  (Fig.  4)  be  taken  proportional 

to  t,  and  BC  to  v.    Then,  since  (19) 

s  =  vt,  and  area  of  rectangle  —  BCAB, 

the  space  is  proportional  to,  or,  in  other  words,  is  repre- 
sented by,  the  rectangle. 

This  principle,  which  is  of  interest  chiefly  from  the 
part  it  takes  in  a  subsequent  demonstration  (25),  means 
simply  that  the  relation  of  the  area  of  the  rectangle  to 
its  sides  is  the  same  as  that  of  the  space  in  uniform 
motion  to  the  velocity  and  time. 

EXAMPLES. 
I.  Uniform  Motion  of  Translation  or  Rotation.    Articles  17-21. 

1.  A  body  travels  30  feet  per  second:  How  far  will  it  go  in  a 
day  of  24  hours? 

2.  A  velocity  of  30  miles  per  hour  corresponds  to  a  rate  of  how 
many  feet  per  second  ? 

3.  A  man  walks  uniformly  4  miles  per  hour :  {a)  How  many 
feet  does  he  go  in  a  second?    (p)  How  many  yards  in  a  minute? 

4.  Two  bodies  start  from  the  same  point  in  opposite  directions, 
the  one  moves  at  a  rate  of  11  feet  per  second,  the  other  at  a  rate 
of  15  miles  per  hour:  ia)  What  will  be  the  distance  between  them 
at  the  end  of  8  minutes?    (6)  When  will  they  be  825  feet  apart? 

5.  How  far  will  the  bodies  in  the  preceding  example  be  apart 
at  the  end  of  the  same  time,  if  they  move  in  the  same  direction  ? 

6.  Two  bodies,  starting  from  the  same  point,  move  along  lines 


22.]  VAEIED   MOTIOIS^ — ACCELERATION.  13 

at  right  angles  to  each  other,  the  first  at  the  rate  of  4^  feet  per 
second,  the  second  at  a  rate  of  200  yards  per  minute :  How  far 
will  they  be  apart  at  the  end  of  an  hour? 

7.  Suppose  the  earth  travels  in  its  orbit  600  million  miles  in 
865idays:  What  velocity  has  it,  expressed  in  miles  per  second, 
supposing  that  the  motion  is  uniform? 


8.  "What  is  the  linear  velocity  of  a  point  on  the  equator  due  to 
the  earth's  rotation? — take  the  equatorial  radius  as  4000  miles. 

9.  What  is  the  linear  velocity  of  a  point  on  the  earth  at  latitude 
60°  from  the  same  cause? 

10.  If  the  linear  velocity  of  a  point  at  the  equator,  due  to  the 
earth's  rotation,  is  «),  show  that  the  velocity  at  any  latitude  (0  is 
equal  to  v  cos  I. 

11.  What  is  the  angular  velocity  of  the  earth's  rotation  per 
second? 

12.  (a)  What  is  the  angular  velocity  of  the  fly-wheel  of  an 
engine,  6  feet  in  diameter,  if  it  makes  40  revolutions  in  a  minute? 
(b)  What  is  the  linear  velocity  of  a  point  on  the  circumference? 

13.  (a)  What  is  the  angular  velocity  of  a  buzz-saw,  having  a 
radius  of  2  feet,  if  it  makes  100  revolutions  per  second?  (b)  How 
far  (in  miles)  will  a  point  on  the  circumference  travel  fn  a  work- 
ing day  of  10  hours? 

14.  The  angular  velocity  of  a  wheel  is  ^7t  per  second:  What  ia 
the  linear  velocity  of  points  at  distances  of  (a)  2  feet,  (b)  4  feet 
and  (c)  10  feet  from  the  centre? 


Varied  Motion — Acceleration. 

22.  Varied  Motion.  The  motion  of  a  body  is  said  to 
be  varied,  and  its  velocity  is  called  variable,  if  it  moves 
through  nneqnal  spaces  in  equal  successive  intervals  of 
time.  The  motion  (supposed  to  be  continuous)  is  said 
to  be  uniformly  varied  if  the  velocity  (1)  increases  or 
(2)  decreases  by  the  same  amount  in  equal  successive 
intervals  of  time,  however  small  these  be  taken. 


14  KINEMATICS.  [23 

In  the  first  case  the  motion  is  uniformly  accelerated, 
as  the  motion  of  a  stone  falling  toward  the  earth;  in  the 
second  case  it  is  uniformly  retarded,  as  that  of  a  stone 
thrown  yertically  upward. 

The  velocity  of  a  body,  if  variable,  is  measured  at  any 
instant  by  the  dlstaiice  through  ivhich  the  body  luould 
pass  in  the  following  unit  of  time,  if  the  motion  were  to 
continue  uniformly  through  that  time  at  the  same  rate. 

Thus,  we  speak  of  the  velocity  of  a  railroad  train  as 
being  at  a  certain  instant  25  miles  per  hour,  meaning 
that,  if  the  rate  were  to  be  kept  up  uniformly  for  the 
hour  following,  the  train  would  pass  over  25  miles.  We 
know,  however,  that  the  supposition  will  not  in  fact  be 
realized.  Again,  the  velocity  of  a  falling  body,  at  a 
certain  instant,  may  be  said  to  be  64  feet  per  second; 
and  by  this  is  meant  that,  if  it  should  move  uniformly 
for  the  next  second  at  the  rate  it  has  at  the  instant 
under  consideration,  it  would  pass  over  64  feet.  But  in 
fact  its  velocity  is  constantly  increasing,  and  it  will 
actually  fall  through  a  space  greater  than  64  feet. 

23.  Acceleration.  If  the  motion  of  a  body  is  uni- 
formly accelerated,  the  equal  increment  of  velocity  for 
each  succeeding  unit  of  time — the  second — is  called  the 
acceleratio7i;  it  is  the  rate  of  change  of  velocity. 

For  example,  a  body  falling  freely  from  rest  toward 
the  earth  acquires  a  velocity  of  about  32  feet  per  second 
at  the  end  of  1  second,  at  the  end  of  2  seconds  its  velo- 
city is  (32)  +  32,  of  3  seconds  it  is  (32  +  32)  +  32,  and 
so  on.  In  other  words,  whatever  the  previous  velocity 
it  may  have  at  any  instant,  in  the  second  following  its 
velocity  is  increased  by  about  32  feet.  This  increment 
of  velocity  of  32  feet-per-second  per  second  (as  it  should 


25.]  VARIED   MOTION — ACCELERATION.  15 

be  expressed  in  full)  is  called  the  acceleration  due 
to  gravity,  and  is  denoted  by  the  letter  g.  In  general 
the  acceleration  due  to  the  action  of  any  force  (as  ex- 
plained in  60)  is  expressed  by  the  letter/. 

It  is  explained  in  a  following  Art.  (64,  p.  61)  that  the  value  of 
g  varies  slightly  for  different  points  on  the  earth's  surface,  being 
greatest  at  the  poles  and  decreasing  toward  the  equator,  w^here  its 
value  is  least.  The  value  for  New  York  is  about  32. 16  (some- 
times called  82^).  It  also  diminishes  as  the  distance  from  the  sur- 
face of  the  earth  increases. 

It  is  also  explained  in  article  65,  p.  62,  that  this  acceleration 
due  to  gravity  is  tlie  same,  at  one  place,  for  all  bodies,  whatever 
their  mass;  that  is,  a  bullet  and  a  feather  will  fall  in  the  same  time 
to  tlie  earth  from  a  given  point,  and  acquire  the  same  velocity,  if 
the  resistance  of  the  air  is  eliminated.     (Read  articles  64,  65,  250.) 

If  the  motion  of  a  body  is  uniformly  retarded,  the 
term  acceleration  is  also  employed  to  indicate  the  equal 
loss  of  velocity  for  each  succeeding  second,  but  it  has 
then  a  negative  sign,  as  having  a  direction  opposite  to 
that  of  the  initial  velocity  of  the  body.  This  is  true  of 
a  body  thrown  up  from  the  earth,  or  of  a  body  projected 
on  a  rough  horizontal  plane  and  retarded  by  friction. 

24.  Velocity  acquired  in  Uniformly  Accelerated  Mo- 
tion. Since  the  acceleration  (/)  of  a  body  is  the  incre- 
ment of  velocity  for  each  successive  second,  it  is  clear 
that,  if  the  body  starts  from  rest,  its  velocity  {v)  at  the 
end  of  ^  seconds  will  be  equal  to//.     That  is: 

V  =  ft\  for  a  falling  body  v  —  gt. 

26.  Space  passed  over  in  Uniformly  Accelerated  Mo- 
tion. The  space  passed  over  hy  a  hody,  starting  from  rest 
and  moving  with  uniformly  accelerated  motion,  is  equal 


le 


KINEMATICS. 


[25. 


to  one  half  the  product  of  the  acceleration  into  the  square 
of  the  time. 

s  =  •J/'f ;  for  a  falling  body  s  =  ^gf. 

Let  AB  (Fig.  5)  be  taken  proportional  to  the  time  {t) 
and  BC,  at  right  angles  to  it,  proportional  to  the  velocity 
{v)  acquired  in  this  time,  and  connect  AC;  it  will  be 
shown  that  the  space  passed  oyer  by  the  body  is  repre- 
sented by  the  area  of  the  triangle  ABC, 


Fig,  5. 


First,  it  is  necessary  to  show  that  if  Ah  represents  any 
other  time  {t')  in  terms  of  the  same  unit,  then  the  cor^ 
responding  perpendicular  he  represents  the  velocity  {v') 
acquired  in  this  time.     For  (24) 


Also, 


V 

=  /^,  and  v'  =ft', 

■■■>=-, 

-  r 

BC 
AB 

=  ^^,  but 

BC 
AB~~ 

V 

T 

he 
'  'Ah 

But,  by  supposition.  Ah  represents  f,  hence  he  must 


26.] 


VARIED    MOTION — ACCELEEATIOlSr. 


17 


represent  v'-,  that  is,  the  corresponding  velocity  acquired 
in  this  time. 

Again,  let  the  time  {t)  be  divided  into  any  number  of 
equal  parts  represented  geometrically  by  Ah',  Vl" ,  etc. 
(Fig.  6).  Erect  the  perpendiculars  I'c' ,  l"c" ,  etc.;  by 
the  preceding  paragraph,  these  perpendiculars  will 
represent  geometrically  the  velocities  acquired  at  the 
end  of  these  times  taken  from  the  beginning.  Now 
suppose  (1)  that  the  body  moves  uniformly  for  each  of 
these  portions  of  time  with  the  velocity  it  has  at  the 
beginning  of  that  interval,  and  (2)  with  that  acquired 


at  the  end.  That  is,  on  the  first  supposition,  it  moves 
for  the  time  AV  with  the  velocity  0;  for  the  time  W 
with  the  constant  velocity  l)'c'\  for  the  time  W"  with 
the  velocity  Vc'' ,  and  so  on.  Then,  by  Art.  21,  the 
sum  of  the  interior  rectangles  ^.AV ,  Ve'  i^—Vh" .  Vc'), 
y'e" ,  and  so  on,  will  represent  the  whole  space  passed 
over  on  this  first  supposition. 

On  the  second  supposition  the  body  moves  for  the 
time  AV  with  the  constant  velocity  l'c'\  for  the  time 
W  with  the  velocity  V'c"\  for  the  time  V'V"  with 
the  velocity  V'c'" ,  etc.     Then,  in  this  case,  the  total 


18  KINEMATICS.  [26. 

space  passed  over  will  be  the  sum  of  the  exterior  rect- 
angles d'h'  {=  Ab'  X  b'c'),  d''b':,  d'"h"',  and  so  on. 

It  is  obvious  that  the  space  represented  by  the  sum  of 
the  interior  rectangles  is  less  than  the  true  space  passed 
over  by  the  body,  and  that  represented  by  the  sum  of  the 
exterior  rectangles  is  greater  than  the  true  space;  and 
each  differs,  by  a  series  of  small  step-like  triangles,  from 
the  area  of  the  triangle.  Now  if  the  number  of  intervals 
into  which  t  is  divided  be  increased  indefinitely,  and 
consequently  the  length  of  each  be  indefinitely  dimin- 
ished, and  the  same  construction  as  that  above  supposed 
be  carried  through,  then  the  sums  of  the  interior  and 
exterior  rectangles  will  approach  the  area  of  the  triangle 
as  their  limit.  But  the  spaces  passed  over  by  the  body, 
upon  the  two  suppositions  made,  also  approach  the 
true  space  (corresponding  to  a  continual  and  unbroken 
increase  in  velocity)  as  their  limit.  But  when  two  sets 
of  variable  quantities,  which  are  always  equal,  simul- 
taneously approach  their  limits,  these  limits  are  equal. 
Therefore 

The  true  space  is  geometrically  represented  hy  the  area 
of  the  triangle. 

r,  s  =  iBC.AB  =  ivt  =  ift\ 

26.  Average  Velocity.     From  the  preceding  article 

s  =  ivt,  and  ^v  =  -.     This  value  of  the  velocity,  ob- 
t 

tained  by  dividing  the  whole  distance  by  the  time,  is 

called  the  average  velocity.     In  the  case  of  uniformly 

accelerated  motion,  the  average  velocity  is  equal  to  one 

half  the  final  velocity  acquired  ;    or,  in   other  words, 

the  space  passed  over  is  equal  to  one  half  the  product  of 

this  final  velocity  into  the  time.     This  is  represented 


27.] 


VARIED    MOTION — ACCELERATION. 


19 


geometrically  by  Fig.  7,  where,  if  BC—^BE,  it  is  seen 
that  the  areas  of  the  triangle  ABC  {=  i.vt)  and  of  the 
rectangle  A  BED  {=  ^v.t)  are  equal. 

The  term  average  velocity  is  also  employed,  in  the  case 
of  varied  motion  in  general,  to  denote  the  result  ob- 
tained by  dividing  the  whole  space  by  the  time.  For 
example,  if  a  train  traverses  100  miles  in  4  hours,  its 
average  velocity  is  said  to  be  J-p  =  25  miles  per  hour, 

c 


ji 


io 


t 

Fig.  7. 


although  its  actual  velocity  may  have  varied  through 
very  wide  limits  during  the  time. 

27.  Formulas  for  Accelerated  Motion.     The  results  of 
articles  24  and  25  give 

V  =  ft;  for  a  falling  body  v  =  gt.  (1) 

s  =  iff;  "        -         -     s  =  igt\         (2) 

Therefore,  eliminating  t  from  (2), 


V  y  ;  for  a  falling  body  2^'  J- .       (3) 

or    v'  =  2/5  )  v""  =  %gs 

These  three  equations  give  the  most  important  rela- 
tions for  bodies  starting  from  rest  and  moving  with  uni- 
formly accelerated  motion.     From  them  we  see  that 


20  KINEMATICS.  [28. 

(1)  The  velocity  acquired  is  proportional  to  the  time. 

(2)  The  space  is  proportional  to  the  square  of  the  time. 

(3)  The  space  is  proportional  to  the  square  of  the  velo- 
city acquired. 

28.  From  equation  (2)  of  the  preceding  article  it  is 
seen  that  the  space  described  in  the  first  second  from  rest 
is  equal  to  one  half  the  acceleration.  Also,  the  spaces 
described  in  1,  2,  3,  4,  etc.,  seconds  are,  by  the  formula: 

1  sec.  2.  3.  4. 

if,        iA        I/,        W>    etc. 

Therefore  the  spaces  described  in  the  first,  second,  third, 
etc.,  seconds  will  be: 

1st  sec.  2d.  3d.  4th. 

if,         if,         if,         If,      etc. 

In  other  words,  the  spaces  described  in  the  successive 
seconds  are  proportional  to  the  numbers  1,  3,  5,  7,  9, 
etc. ;  for  the  7^*^  second  the  space  will  be  by  this  law 

— ^r — /,  or  for  a  falling  body  — - — g\  this  is  equal  to 

the  space  passed  over  in  {n  —  1)  seconds  [=  i/(^^ — l)'^] 
subtracted  from  the  space  passed  over  in  n  seconds 

EXAMPLES. 

II.    Uniformly  Accelerated  Motion  (Articles  22-28).     A.  Falling 
Bodies  {take  g  =  32). 

[It  is  to  be  understood  in  each  case  that  the  body  falls  from 
rest,  and  that  the  resistance  of  the  air  is  neglected.  It  is  to  be 
remembered,  also,  that  the  assumption  that  the  value  of  g  is  con- 
stant for  points  above  the  surface  of  the  earth  is  not  rigidly  true.] 

1.  A  body  falls  15  seconds:  Required  (a)  the  velocity  acquired; 


28.]  VAEIED    MOTION — ACCELERATION.  21 

(b)  the  whole  distance  fallen  through ;  (c)  the  space  passed  over  in 
the  last  second  of  its  fall;  (d)  the  space  in  the  last  three  seconds. 

2.  A  body  has  fallen  through  5184  feet:  Required  (a)  the  time 
of  falling ;  (b)  the  final  velocity. 

3.  A  body  has  acquired  in  falling  a  velocity  of  513  feet  per 
second:  Required  (<)^)  the  time  of  falling;  (5)  the  distance  fallen 
through. 

4.  A  body  in  falling  passed  over  336  feet  in  the  last  second: 
Required  (a)  the  time  of  falling;  (b)  the  distance  fallen. 

5.  A  body  in  falling  passed  over  1008  feet  in  the  last  three 
seconds:  Required  (a)  the  time  of  falling;  (b)  the^  distance  fallen 
through. 

6.  What  is  the  ratio  of  the  velocities  of  a  falling  body  at  the  end 
of  the  first  i,  i,  1,  3,  and  4^  seconds  ?  Find  the  actual  veloci- 
ties in  this  way,  from  the  velocity  at  the  end  of  1  second  (g). 

7.  What  is  the  ratio  of  the  spaces  passed  over  by  a  falling  body 
in  i,  i,l,  3,  4^  seconds  ?  Obtain  the  respective  distances  in  this 
way,  from  that  of  1  second  (16  feet). 

8.  A  sand-bag  is  dropped  from  a  balloon,  which  is  for  the 
moment  at  rest  at  a  height  of  3  miles :  Required  {a)  the  time  of 
falling  to  the  earth,  and  (b)  the  velocity  acquired. 

9.  What  is  the  distance  fallen  through  in  the  third  of  a  second, 
commencing  (a)  the  6th  second,  and  (b)  the  11th  second? 

10.  Two  balls  are  dropped  at  the  same  instant  from  points  100 
feet  apart  vertically:  What  distance  will  separate  them  at  the  end 
of  2,  3,  and  5  seconds  ? 

11.  Two  balls  A  and  B  are  dropped  from  a  height,  B  2  seconds 
after  the  other  :  (a)  How  far  apart  will  they  be  after  B  has  fallen 
2,  3,  and  5  seconds  ?   (b)  When  will  they  be  416  feet  apart  ? 

12.  A  stone  is  dropped  down  a  well  224  feet  deep:  How  soon 
will  the  splash  in  the  water  be  heard  at  the  top  if  the  velocity  of 
the  sound  is  1120  feet  per  second  (corresponding  to  a  tempera- 
ture of  the  air  of  about  60°  F.)? 

13.  A  stone  is  dropped  from  the  top  of  a  cliff,  and  after  6^ 
seconds  it  is  heard  to  strike  the  ground  below :  How  high  is  the 
cliff,  taking  the  velocity  of  sound  as  1152  feet  per  second  (tei 
perature  of  air  about  90°  F.)? 


22  KINEMATICS.  [29 


B.  General  Case. — Acceleration  =f. 
[The  motion  is  assumed  to  be  uniformly  accelerated.] 

1.  A  body  moves  100  feet  in  the  first  5  seconds  from  rest. 
What  is  the  acceleration  ? 

2.  A  body  moves  10  feet  in  the  first  second:  (a)  "What  is  the 
acceleration  ?  (b)  How  far  will  it  go  in  8  seconds  ?  (c)  What  wil\ 
be  its  final  velocity  at  the  end  of  this  time  ? 

3.  The  acceleration  is  12  f eet-per-second  per  second :  (a)  What 
velocity  does  a  body  acquire  in  6  seconds  ?  (b)  What  space  does  it 
pass  over  ? 

4.  A  body  passes  over  36  feet  in  the  fifth  second :  What  is  th( 
acceleration  ? 

5.  The  acceleration  due  to  the  attraction  of  Jupiter  for  bodies 
on  or  near  its  surface  is  about  2.6  times  ^r:  (a)  Wnai  velocity 
would  a  falling  body  acquire  in  3  seconds  ?  (b)  What  spnce  would 
it  pass  through  in  this  time  ? 

6.  What  time  would  be  required  in  the  above  case  (5)  for  a  body 
to  fall  2340  feet  ? 

7.  The  acceleration  of  gravity  on  the  moon  is  about  Ig  :  How 
long  and  how  far  must  a  body  fall  to  acquire  a  velocity  of  32  feet? 

8.  The  acceleration  of  gravity  on  the  sun  is  about  28  X  fi'  • 
Compare  the  acquired  velocities  and  spaces  fallen  through  for 
the  first  three  seconds  with  those  true  for  the  earth. 

9.  A  body  moves  45  feet  in  3  seconds,  and  80  feet  in  the  next 
2  seconds :  Is  its  motion  uniformly  accelerated  ? 

10.  A  body  passes  over  50  feet  in  5  seconds:  What  distance 
must  it  go  in  the  next  5  to  satisfy  the  condition  of  uniformly 
accelerated  motion  ? 

Composition  and  Reoolutmi  of  Velocities — Uniform 
Motion. 

29.  Composition  of  Motions  in  General.  It  was  ex- 
plained in  Art.  11  that,  when  a  body  is  said  to  be  in 
motion,  reference  is  always  made  to  some  other  body 
with  respect  to  which  the  first  body  changes  its  posi- 


81.]  COMPOSITION   OF   MOTIONS.  23 

tion.  In  many  cases  which  arise  we  have  to  consider 
not  the  simple  motions  of  bodies,  but  their  actual 
motions  as  composed  of  several  different  motions.  For 
example,  a  man  walking  on  the  deck  of  a  steamboat  is 
in  motion  with  reference  to  it,  but  the  boat  in  turn  is 
in  motion  as  compared  with  the  neighboring  shore. 
Therefore  his  actual  motion  with  reference  to  the  land 
is  composed  of  his  own  independent  motion  and  that  of 
the  boat.  Hence  we  have,  in  such  cases,  to  do  with  the 
coexistence  of  motions;  and  the  problem  arises,  when  the 
separate  motions  are  given,  to  find  the  actual  resulting 
motion  in  rate  (velocity)  and  direction. 

30.  Resultant  and  Component  Velocities.  When  a 
body  tends  to  move  at  the  same  time  with  several 
different  velocities,  either  in  the  same  or  different 
directions,  the  actual  velocity  due  to  the  combination 
of  all  is  called  the  resultant,  and  the  separate  velocities 
are  called  the  components.  The  process  of  finding  the 
resultant,  when  the  components  are  given,  is  called  the 
Composition  of  Velocities. 

31.  Composition  of  Constant  Velocities  in  the  same 
Straight  Line.  The  resultant  of  two  component  velocities 
in  the  same  direction  is  equal  to  their  sum;  if  they  have 
opposite  directions,  it  is  equal  to  their  difference.  In 
general,  if  of  several  velocities  those  in  one  direction  are 
called  plus  (+),  and  those  in  the  opposite  are  called 
minus  (—),  the  resultant  is  equal  to  their  algebraic  sum. 

For  example,  a  boat,  moving  uniformly  at  the  rate  of 
6  miles  per  hour  down  a  stream  running  at  the  uniform 
rate  of  4  miles,  has  a  resultant  velocity  in  the  same 
direction  of  10  miles  (6  +  4).  If  the  boat  is  headed  up 
stream  and  keeps  the  same  rate,  the  resultant  velocity  is 


24  KINEMATICS.  [32. 

also  up  stream  and  equal  to  2  miles  (6  —  4).  If,  iu  the 
latter  case,  the  stream  had  a  velocity  of  8  miles,  the 
resultant  velocity  would  be  equal  to  —2,(6—8);  that 
is,  the  boat  would  in  fact  drift  down  stream  at  this  rate. 

32.  Composition  of  two  Constant  Velocities  not  in  the 
same  Straight  Line.  If  the  two  component  velocities  are 
not  in  the  same  straight  line,  then  the  resultant  velocity 
will  lie  between  them,  and  will  be  determined  in  direc- 
tion and  magnitude  by  the  Parallelogram  of  Velocities. 

33.  Parallelogram  of  Velocities.  This  principle  is 
stated  as  follows  :  If  the  component  velocities  he  repre- 
sented in  direction  and  magnitude  hy  the  two  adjacent 
sides  of  a  parallelogram,  the  resultant  velocity  will  be 
given  by  the  diagonal  passing  through  their  point  of 
intersection.     Suppose  a  body  tends  to  move  uniformly 


from  A  toward  B  (Fig.  8)  with  a  velocity  u,  represented 
by  the  line  AB;  also  at  the  same  instant  from  A  to  D 
with  a  velocity  v,  represented  by  AD,  then  the  body  will 
actually  move  in  the  direction  A  C  with  a  constant  velo- 
city represented  by  AC.      (See  Arts.  68,  b,  and  128.) 

For  the  body,  if  it  had  only  the  velocity  u,  would  in 
one  second  move  from  A  to  B,  and  if  only  the  velocity 
V,  would  move  from  A  to  D;  but  the  motion  in  the  one 


84.]  COMPOSITION   OF  MOTIONS.  2o 

direction  cannot  effect  that  in  the  other  if  they  go  on 
together,  so  that  at  the  end  of  the  given  time  the  bod}i 
will  actually  be  at  C,  having  moved  along  the  straight 
line  A C.  But,  again,  liAb',  Vh",  V'l"\  etc.,  be  taken 
to  represent  the  motion  of  the  body  in  equal  intervals  of 
time  in  the  direction  AB  with  the  velocity  w,  and  Ad' , 
d'd'\  d"d"' ,  etc.,  the  motion  in  the  same  intervals  of 
time  in  the  direction  AD  with  the  velocity  v,  the  result- 
ant motion  will  be  represented  by  Ac' ,  c'c",  c"c"',  etc. 
But  these  distances  are  equal,  since  they  are  by  similar 
triangles  proportional  to  the  equal  distances  Ah',  W , 
etc.  (or  Ad' ,  d'd" ,  etc.);  therefore  the  motion  of  the 
body  in  the  resultant  direction  is  also  uniform. 

34.  Calculation  of  the  Magnitude  and  Direction  of  the 
Resultant  Velocity.  From  trigonometry  we  have  (see 
also  Art.  130) 

^(7»  =  AE"  +  AB"  +  2AB.AD  cos  y. 
Therefore 

F""  =  w''  +  ^^  +  ^^^  cos  y. 

jd.  ^ 


Also,  in  the  triangle  ABC  (Fig.  9)  the  component 
velocities  are  represented  by  ^^  =  w,  and  BC{=  AD)  = 
v\  the  resultant  velocity  is  AC  ^V\  also,  BA C  =  o', 
ACB  {=  CAD)  =  A  and  CBA  =  180°  -  BAD  =  180° 
—  y  —  180°  —  {a  -\-  ft).    Hence  the  relations  in  direc- 


26 


KINEMATICS. 


[35. 


tion  and  magnitude  of  the  resultant  and  component 
velocities  may  be  calculated  by  the  usual  methods  for 
the  solution  of  this  plane  triangle,  where  three  parts 
are  given  and  the  others  required. 

It  is  further  seen  from  this  case  that  the  relation  of 
the  two  component  velocities  and  their  resultant  may  be 
expressed  geometrically  by  the  triangle  ABC,  hence 
sometimes  called  the  Triangle  of  Velocities. 

36.  For  the  special  case  (Fig.  10)  where  the  directions 


of  the  component  velocities  are  at  right  angles  to  each 
other,  the  relations  are  more  simple.     Here 


V=  4/^;^  -|-  ^2-    also,    cos  a:  =  sin  /?  = 


V 


sin  «  =  cos^  = 


V 


tan  Of  =  cot/? 


36.  The  following  cases  may  be  taken  as  illustrations  of  the 
Parallelogram  of  Velocities. — Suppose  a  boat  to  move  uniformly 
at  the  rate  of  u  miles  per  hour  across  a  stream  running  at  the  rate 
of  xi  miles  per  hour.  Here  the  velocities  u  and  «  are  the  com- 
ponents, and,  if  Ah,  Ad  be  taken  to  represent  them  in  their 
respective  directions  (Fig.  11),  the  resultant  velocity  will  be  given 


37.] 


COMPOSITION   OF  MOTIONS. 


27 


in  directiou  and  magnitude  by  the  diagonal  Ac.  Hence  the  boat 
will  actually  move  in  the  direction -4.c,  I  tan  a  =  —I  at  the  rate  of 

V  miles  an  hour  (=  Vv'^-\-  u%  and  will  reach  C  in  the  same  time 
in  which  it  would  have  gone  without  the  current  directly  across 
to  B,  or  would  have  drifted  with  the  current  to  D. 

Again,  suppose  the  component  velocities  as  above;  if  it  be 
required  that  the  boat  shall  go  directly  across  to  C,  here  the  direc- 
tion of  the  resultant  is  AG  (Fig.  12),  and  the  components  are 


::3 


/..•• 


,^- 


Fio.  11. 


Fio.  12. 


Ab{=:u)  and  Ad  (=  v).  Hence  the  boat  must  be  headed  up 
stream  at  an  angle  BAGAsin  a  =  —\,  and  the  resultant  velocity 
will  be  expressed  by  Vu^  —  'v^. 

37.  Composition  of  several  Constant  Velocities.     If 

there  are  more  than  two  component  velocities,  then  the 
method  of  finding  their  resultant  is  as  follows:  Find  the 
resultant  of  two  of  the  component  velocities,  then  of  this 
resultant  and  the  third  component,  again  of  the  last 
resultant  and  the  fourth  component,  and  so  on.  Thus 
let  AB,  AC,  AD,  AE  (Fig.  13),  represent  several  com- 
ponent velocities;  the  resultant  of  AB  and  ^6'  is  (33) 
the  diagonal  Ac;  again,  the  resultant  of  Ac  and  AD,  that 


28  KINEMATICS.  [38. 

is  of  AB,  AC,  AD,  is  Ad;  still  again,  the  resultant  of 
Ad  and  AE,  that  is  of  the  four  original  velocities,  is  Ae. 
It  will  be  seen  by  comparing  Figs.  13  and  14,  that  the 
sides  of  the  polygon  a  b  c  d  e  (Fig.  14)  represent  the 
four  velocities  and  their  resultant.  Hence,  in  general, 
if  the  component  velocities  be  laid  off  in  order  of  direc- 
tion, as  ab,  be,  cd,  de  (Fig.  14),  the  side  which  com- 


FiQ.  13.  Fig.  14. 

pletes  the  polygon  so  formed,  viz.  ae,  represents  the 
resultant  velocity.  This  is  sometimes  called  the  Poly- 
gon of  Velocities. 

38.  Resolution  of  Velocities.  The  process  of  finding 
the  component  velocities,  which  shall  be  equivalent  to  a 
given  resultant  velocity,  is  called  the  Resolution  of  Velo- 
cities. If  the  required  components  are  two  in  number 
and  their  directions  are  given,  then  their  magnitude  is 
found  by  completing  the  parallelogram  whose  diagonal 
is  the  given  resultant  velocity  and  whose  sides  have  the 
given  directions. 

Let  (Fig  15)  A  C  represent  the  resultant  velocity,  and 
let  ^Xand  ^l^be  the  directions  of  the  required  com- 
ponents; draw  through  G  the  lines  CD,  CB  parallel 
respectively  to  AX  and  AY,  then  AB  and  AD,  the 


88.] 


COMPOSITION    OF  MOTIONS. 


29 


sides  of  the  parallelogram  thus  formed,  will  represent 
the  required  components  in  direction  and  magnitude. 

Yl  T 


J) . 


\B 


Fia.  15. 


Fio.  16. 


If  the  directions  AX  and  AY  are  at  right  angles 
(Fig.  16),  and  a  represents  the  angle  CAB,  then  the 
components  are  AB  =  AC  oos  a  and  AD  =  AC  sin  a. 

For  example,  suppose  a  boat  to  move  uniformly  in  the 
direction  AC  (Fig.  17)  in  virtue  of  its  own   motion 


.S 


Fig.  17. 


directly  across  the  stream  and  the  velocity  of  the  cur- 
rent toward  D,  taken  together,  both  being  uniform. 
Then  it  Ac  represents  this  resultant  velocity,  the  com- 
ponent velocities  will  be  given  hj  Ab  {=  Ac  cos  a)  and 
Ad  (=  Ac  sin  a).  If  the  transfer  across  the  stream  were 
alone  desired,  the  component  Ab  in  this  direction  might 
be  called  the  effective  velocity. 


30  KINEMATICS.  [38. 


EXAMPLES. 

III.  Composition  of  Velocities.    Articles  29-37. 

[The  velocities  are  supposed  to  be  constant  in  all  cases.] 

1.  The  velocity  of  a  steamboat  is  5  miles  per  hour,  that  of  the 
stream  is  4  miles,  and  a  man  walks  the  deck  from  stern  to  bow  at 
the  rate  of  3  miles :  Required  the  actual  velocity  of  the  boat  (a)  if 
headed  up  stream,  and  ib)  down  stream ;  also  (c,  d),  that  of  the  man 
in  each  case. 

2.  The  velocities  of  boat  and  stream  are  as  in  example  1,  and 
the  boat  is  headed  directly  across  the  stream  (Fi^.  11):  (a)  What 
will  be  the  actual  direction  of  the  boat's  motion  ?  (5)  What  the 
rate  of  its  motion  ?  {c)  How  long  will  the  passage  take  if  the 
stream  is  2  miles  wide  ?    {d)  Where  will  the  boat  land  {BG  =  ?). 

3.  The  velocities  of  boat  and  stream  are  as  in  1  and  2,  but  it  is 
required  that  the  boat  shall  go  directly  across  from  ^  to  C  (Fig. 
12):  {a)  In  what  direction  must  the  boat  be  headed  ?  (5)  What 
will  be  its  actual  velocity  across  ?  (c)  What  will  be  the  time  of 
passage,  the  width  being  2  miles  ? 

4.  Find  answers  for  the  three  questions  in  example  3,  on  the 
supposition  that  the  boat  is  to  reach  a  point  30°  up  stream  from 
the  starting-point. 

5.  Find  answers  for  the  three  questions  in  example  3,  on  the 
supposition  that  the  boat  must  reach  a  point  30°  down  stream. 

6.  The  velocity  of  the  boat  is  4  miles  per  hour,  and  that  of  the 
stream  5  miles;  the  width  of  the  stream  is  1  mile:  What  is  nearest 
point,  to  that  directly  across  from  the  starting-point,  which  the 
boat  can  reach?  (Solve  this  problem  by  geometrical  construction.) 

7.  A  ball  on  a  horizontal  surface  tends  to  move  north  with  a 
velocity  of  12  feet  per  second,  and  east  with  a  velocity  of  5  feet 
per  second:  {a)  What  will  be  the  actual  velocity,  and  (p)  in  what 
direction  ? 

8.  A  ball,  moving  north  at  a  rate  of  8  feet  per  second,  receives 
an  impulse  tending  to  make  it  move  due  north-east  with  the  same 
velocity:  {a)  What  path  will  it  take,  and  (5)  at  what  rate  will  it 
move  ? 

9.  A  man,  skating  uniformly  at  a  rate  of  12  feet  per  second. 


38.]  COMPOSITION   OF   MOTIONS.  31 

projects  a  ball  on  the  ice  in  a  direction  at  right  angles  to  his 
motion  at  a  rate  of  9  feet  per  second :  What  is  (a)  the  actual  rate, 
and  (b)  the  direction  of  its  motion  (friction  neglected)? 

10.  A  ball  tends  to  move  north  at  the  rate  of  8  feet  per  second, 
also  S.  60°  E.  and  S.  60"  W.,  each  at  the  same  rate:  What  is  its 
actual  velocity  ? 

11.  A  ball  tends  to  move  east  5  miles  per  hour,  also  N.  45°  W. 
and  S.  45°  W.,  each  at  the  same  rate:  Required  the  direction  and 
rate  of  motion. 

12.  If  a  boat  headed  directly  across  a  stream  moves  at  the  uni- 
form rate  of  100  yards  a  minute,  while  the  current  runs  80  yards 
a  minute:  (a)  In  what  direction  will  it  actually  go,  and  (b)  what 
distance  will  it  land  down  stream  ?  (c)  What  should  be  its  course 
in  order  that  it  may  reach  a  landing-place  directly  opposite  the 
starting-point,  and  (d)  how  long  would  the  passage  take  ?  The 
width  of  the  stream  is  1200  yards. 

IV.  Resolution  of  Constant  Velocities.    Article  38. 

1.  A  ball  tends  to  move  in  a  certain  direction  at  a  rate  of  9  feet 
per  second,  but  it  is  constrained  to  move  at  an  angle  of  30°  with 
this  direction :  Required  its  velocity  in  the  latter  direction.  (Fig.  16.) 

2.  A  body  moves  uniformly  about  a  semi-circumference  at  the 
rate  of  12  feet  per  second:  What  is  the  component  of  its  velocity 
parallel  to  the  diameter  when  it  is  30°,  60°,  90°,  120°,  and  180° 
from  the  starting-point?  (Fig.  IG) 

3.  A  ball  rolls  at  the  rate  of  8  feet  per  second  across  the  diagonal 
of  a  rectangular  room  ABCD  whose  dimensions  are  15  X  20 
(=  AB  X  AG)\  What  is  its  rate  of  motion  parallel  to  each  side? 

4.  A  body  moves  N.  30°  E.  at  a  rate  of  6  miles  per  hour; 
Required  its  rate  of  motion  northerly  and  easterly? 

5.  A  boat,  though  headed  directly  across  a  stream,  actually 
moves  diagonally  across  the  stream  at  an  angle  of  30°  {BAG,  Fig, 
11, 'p.  27),  and  at  a  rate  of  10  miles  per  hour:  Required  {a)  the  rate 
of  the  boat,  and  (p)  of  the  current,  each  taken  independently. 

6.  A  boat  steams  directly  across  a  stream  1800  yards  wide  in 
30  minutes,  the  current  flowing  all  the  time  at  the  rate  of  80  yards 
per  minute :  What  would  be  the  direction  and  rate  of  motion  if 
there  were  no  current  ? 


32 


KINEMATICS. 


[89. 


7.  A  balloon  has  a  velocity  of  20  feet  per  second  in  an  upward 
direction  which  makes  an  angle  a  with  a  vertical  line :  If  its  ve- 
locity vertically  upward  would  be  1000  feet  per  minute,  what  is 
its  horizontal  velocity  due  to  the  wind  ?    What  is  a  ? 


Composition  and  Resolution  of  Accelerations, 

39.  Composition  and  Resolution  of  Accelerations.   The 

composition  and  resolution  of  velocities  may  be  extended 
also  to  the  case  of  uniform  accelerations,  the  method 
being  in  all  respects  similar  to  that  in  the  preceding 
articles.  The  sides  of  the  parallelogram  here  represent 
the  component  accelerations,  and  the  diagonal  the  re- 
sultant acceleration. 

The  simplest  application  of  the  principle  of  the  reso- 
lution of  accelerations  is  to  the  case  of  motion  down  an 
inclined  plane  (40). 

40.  Motion  down  an  Inclined  Plane.  The  direction  of 
the  acceleration  of  gravity  is  that  of  a  vertical  line,  and 


Fio.  18. 

a  body  falls  in  this  direction  if  entirely  free;  but  a  body 
on  an  inclined  plane  is  only  free  to  slide  along  it,  and  the 
acceleration  is  here  that  component  of  the  whole  accele- 
ration which  is  parallel  to  the  plane;  viz.,  g  sin  a. 

Let  (Fig.  18)  ac  be  taken  to  represent  the  vertical 
acceleration  g\  the  directions  of  its  components  are 
respectively  parallel  and  perpendicular  to  the  plane,  and 


40.]  COMPOSITION   OF  MOTIONS.  33 

are  represented  by  ah  and  ad.  But  lac  —  HLK  —  a, 
and  therefore  ad  —  ic  =  ac  sin  a.  That  is,  ady  or  the 
acceleration  down  the  plane,  is  equal  to  g  sin  a. 

The  formulas  of  Art.  27,  for  a  falling  body,  are  then 
applicable  to  the  case  of  a  body  sliding  down  a  smooth 
inclined  plane,  if  for  g  we  write  g  sin  a.     That  is  : 

V  =  g  sin  a.  t, 
s  =  ig  sin  a .  f, 
v^  =  2g  sin  a .  s. 

In  the  last  formula,  if  the  body  descends  from  ^to  L, 
s  =  HL,  and  5  sin  «;  =  UK  or  h,  the  height  of  the 
plane; 

From  this  equation  it  follows  that:  the  velocity  acquired 
in  descendiyig  any  inclined  plane  is  the  same  as  that 
gained  in  falling  through  the  vertical  height  of  the  plane. 
This  is  also  true  for  a  continuous  curve.  This  principle 
finds  an  application  in  Art.  244. 

EXAMPLES. 

V.  Falling  down  an  Inclined  Plane.     Article  40. 

[The  plane  is  supposed  to  be  perfectly  smooth,  so  that  there  is  no 
friction.] 

1.  The  angle  of  the  plane  is  30":  Required  {a)  the  acceleration 
down  the  plane;  (J))  the  distance  fallen  through  in  4  seconds;  (c) 
the  velocity  acquired;  {d)  the  distance  in  the  last  second. 

2.  The  height  of  the  plane  is  100  feet  and  the  length  400:  {a) 
"What  is  the  time  required  to  reach  the  bottom  ?  (p)  What  is  the 
velocity  acquired  ? 

8.  The  angle  of  the  plane  is  45° :  Required  the  time  of  falling 
144  feet. 


34  KINEMATICS.  [41. 

4.  The  length  of  a  plane  is  576  feet,  a  body  falls  down  it  in  24 
seconds:  {a)  What  is  the  acceleration  ?  (6)  What  is  the  height  of 
the  plane  ? 

5.  The  height  of  a  plane  is  98  feet,  and  a  body  gains  a  velocity 
of  20  feet  per  second  in  falling  5  seconds  on  it:  Required  {a)  the 
acceleration;  (5)  the  length  of  the  plane. 

6.  The  height  of  a  plane  is  256  feet,  a  body  reaches  the 
bottom  in  16  seconds:  {a)  What  is  the  length  of  the  plane  ?  (6) 
What  is  the  velocity  acquired  ? 

7.  Several  planes,  having  the  same  altitude,  viz.  400  feet,  have 
lengths  600,  800,  1200,  and  1600  feet:  Compare  the  times  of 
descent  and  acquired  velocities  for  each. 

8.  Show  that  for  several  planes  having  the  same  altitude  the 
times   of    descent   are   proportional   to   the   lengths ;    that  is, 

t         I         ^      , 

9.  Prove  that  the  time  of  falling  from  rest  down  a  chord  of  a 
vertical  circle,  drawn  from  the  highest  point,  is  constant. 


Composition  of  Uniform  and  Accelerated  Motion  in  the 
same  Line. 

41.  Composition  of  Uniform  and  Accelerated  Motion  in 
the  same  Straight  Line.  (Only  the  cases  of  uniformly 
accelerated  and  retarded  motion  will  be  considered.) 

If  of  two  component  velocities  one  is  constant  (u) 
and  the  other  is  uniformly  increasing — that  is,  tending 
to  produce  uniformly  accelerated  motion — but  both  in 
the  same  line,  then  the  resultant  is  equal  to  their  sum 
or  difference  according  as  they  have  the  same  (a)  or 
opposite  directions  (b). 

(«)  In  the  first  case,  represent  the  uniform  velocity 
by  w,  and  that  produced  by  the  accelerated  motion  by 

V  (=  ft);  then,  if  Fis  the  resultant  velocity, 

V  =  u  -{-  V  =  u  +ft;  tor  Si  falling  body  V=u  +  gf,    (1) 


42.]  COMPOSITION   OF   MOTIONS.  35 

The  last  formula  applies  to  the  case  of  a  body  pro- 
jected \fitli  an  initial  velocity  vertically  downward  toward 
the  surface  of  the  earth  from  a  point  above.  The 
resultant  velocity  is  the  sum  of  this  initial  velocity  and 
that  due  to  its  accelerated  motion  caused  by  gravity  (24). 

{b)  In  the  second  case 

V  =  u  —  V  =  u  —  ft;  iov  a,  falling  body  V  =u  —gt,    (2) 

The  last  formula  here  applies  to  the  case  of  a  body 
projected  vertically  upward  from  the  earth  with  an 
initial  velocity  u;  its  resultant  velocity  at  any  moment 
is  then  equal  to  the  initial  velocity  diminished  by  the 
velocity  due  to  the  accelerated  motion  downward  (that 
is,  in  the  opposite  direction)  caused  by  gravity. 

42.  The  distance  (s)  which  a  body  passes  over  in  a 
given  time,  in  the  above  examples,  is  to  be  found  by  tak- 
ing the  sum,  in  the  first  case,  and  the  difference,  in  the 
second,  of  the  space  that  would  be  passed  over  if  it 
moved  uniformly  for  the  time  t  (19)  with  the  velocity  u, 
and  that  it  would  pass  over  independently  in  the  same 
time  in  consequence  of  the  accelerated  motion  (27). 

Therefore  (a) 

s  =  tit-{-  iff;  for  a  falling  body  s  =  ut  -{-  igf,    (3) 
And  {i) 

s  —ut  —  iff;  for  a  falling  body  s  =  ut  —  igf.    (4) 
By  combining  equations  (1)  and  (3),  since  V=  u  -\-  ft, 
V'=:u'-{-  2uft  +  fH'=u'  +  2f{ut  +  i  ft')  =  tf-\-2fs. 
Therefore 

V'  =  u'-{-2fs;  for  a  falling  body  V'=u' -\-  2gs .     (5) 
In  the  same  manner 
V  =  if  -  2fs ;  for  a  falling  body  V"  =  u"  -  2gs.     (6) 


36 


KINEMATICS. 


[43. 


ilM^ 


43.  Geometrical  Representation.  It  was  shown,  in 
Art.  21,  that  the  space  passed  over  by  a  body  moving 
uniformly  may  be  represented  geometrically  by  a  rect- 
angle; and  again,  in  Art.  25,  that  the  space  described 
by  a  body  moving  with  uniformly  accelerated  motion 
may  be  represented  by  a  right-angled  triangle.  If  now 
a  body  has  an  initial  velocity  in  the  same  or  opposite 
direction  to  that  in  which  it  begins  to  move  with  uni- 
formly accelerated  motion,  the  space  passed  over  will  be 
represented  by  a  geometrical  figure  formed  by  the  com- 
bination of  the  rectangle  and  triangle. 

For  example,  in  Fig.  19,  let  AB  represent  the  time 
(t),  BC  the  initial  velocity  (^i),  also  CE  the  velocity  {v) 
acquired  in  this  time  and  in  the  same  direction  as  u\ 
then  will  the  whole  space  passed  over  be  represented  by 
the  figure  ABED,  which  is  the  sum  of  the  rectangle 
ABCD  (ut)  and  the  triangle  DCE  {ivt  =  iff). 


Again,  suppose  the  accelerated  motion  to  be  in  a 
direction  opposite  to  that  of  the  initial  velocity.  Let 
AB  (Fig.  20)  represent  the  time  (t),  and  BC  the  initial 
velocity  (it),  also  take  CE  to  represent  the  velocity 
acquired  (v)  in  the  given  time;  then  the  space  described 
will   be  proportional  to  the   area   of   the  quadrilateral 


44.]  COMPOSITION   OF  MOTIONS.  37 

ABED,  wliicli  is  the  difference  between  the  rectangle 
ABCD  (ut)  and  the  triangle  DCU  {ivt  =  iff). 

44.   Motion  of  a  Body  projected  vertically  upward. 

The  three  formulas  obtained  in  Art.  42,  which  give  the 
relations  of  the  velocity,  space,  and  time  of  a  body  which 
has  an  initial  velocity  in  a  direction  opposite  to  that  in 
which  it  tends  to  move  with  accelerated  motion,  have 
an  especial  importance.     They  are: 

V=u-ft,  s  =  ut-\ft\  V'  =  u'-2fs, 

For  a  falling  body  these  are: 

V=u-  gt,  (1)     s=ut-  igf,  (2)     V  =  w'  -  2gs.  (3) 

The  relations  given  below  are  deduced  from  the  last 
three  equations,  since  the  case  of  the  body  projected 
vertically  upward  is  practically  the  most  important,  but 
all  the  results  obtained  may  be  made  general  by  writing 
/  for  g. 

1.  The  Time  of  Ascent,  From  equation  (1),  if  ^  =  0 
— that  is,  at  the  moment  of  starting —  V  =  u,  the  initial 
velocity;  as  t  increases  V  diminishes,  and  when  gt  =  u, 

or  t  =  -,  then   V  =  0.      That  is,  at  a  time  after  the 

u 
starting,  expressed  by  ^  =  -,  the  body  will  for  an  instant 

come  to  rest. 

2.  Time  of  Descent.     If  in  the  same  equation  gt  is 

greater  than  u,  i.e.  t  is  greater  than  -,  the  value  of  V 

%/ 
will  be  negative;  in  other  words,  the  body  will  begin  to 
descend.     "When  it  reaches  the  starting-point  again, 
5=0,  and  therefore,  from  equation  (2), 

ut  -  igf  =  0,       and      ^  =  0  or  — . 

9 


38  KINEMATICS.  [41 

The  value  t  =  0  corresponds  obyiously  to  the  moment 
of  starting,  and  t  =  —  means  that  at  the  end  of  this 

time  the  body  will  have  returned  to  the  starting-point. 

2u 
The  time  of  ascent  and  descent  is  then  — ;  and  since 

the  former  =  — ,  the  time  of  descending  must  be  also 

equal  to  — . 

3.  Height  of  Ascent.     At  the  highest  point  reached 
F  =  0,  and  therefore  in  equation  (3) 

0  =  ^^'^  —  2(]s,         and        s  =  — — , 

and  this  value  gives  the  distance  ascended. 

This  can  also  be  obtained  from  equation  (2);  for  at 

the  highest  point  t  =  —,  therefore 


9        ^9        ^' 

This  equation  is  the  same  as  (3)  in  Art.  27;  hence  the 
result  here  obtained  may  be  stated  in  this  form :  A  body 
projected  vertically  uj)  will  ascend  to  a  height  from 
which  it  must  fall  to  acquire  a  velocity  equal  to  that  of 
its  projection. 

4.  Velocity  acquired  in  descending.      The  time  rc- 

2w 
quired  for  the  whole  ascent  and  descent  is  — ,  therefore 

in  equation  (1) 

T7  ^^  O 

V  z=z  u  —  a  . z=u  —  2u  =:■   —  U, 


V       f 


46.]  COMPOSITION   OF   MOTIONS.  39 

or  the  velocity  acquired  in  descending  is  equal  to  the 
initial  velocity,  but  in  the  opposite  direction. 
If  in  equation  (3)  we  let  s  =  0,  then 

which  result  corresponds  to  that  just  given. 

Furthermore,  for  any  value  of  s  there  will  be  two 
different  values  of  t  from  equation  (2),  corresponding  to 
the  time  when  it  passes  the  given  point  on  the  ascent,  and 
that  when  it  returns  to  it  on  the  descent.  Also,  at  any 
point  on  the  descent  the  velocity  (equation  3)  will  be 
the  same  with  the  contrary  sign  as  that  on  the  cor- 
responding point  in  the  ascent. 

Aj  /     45.  Projected  up  or  down  an  Inclined  Plane.     For  a 

body  moving  up  or  down  a  smooth  inclined  plane  with 
an  initial  velocity  w,  the  relations  are  the  same  as  those 
given  in  articles  41,  42,  and  44,  except  that,  as  in  40,  in 
every  case  we  must  write  g  sin  a  for  g. 

Down.  Up. 

V  =  u  -\-  g  sin  a.t,  V  =  u  —  g  sin  a.t, 

8  =z  ut  -\-  ig  sin  a.f,  s  =  ut  —  ig  sin  a.f, 

V^  =  u'  -{-  2g  sin  a.s.  V^  =  u^  —  2g  sin  a,s. 

EXAMPLES. 

VI.  Bodies  projected  vertically  doicnward.     Articles  41,  42. 
[The  resistance  of  the  air  is  neglected.] 

1.  A  body  is  thrown  vertically  down  with  an  initial  velocity  of 
36  feet  per  second :  Required  (a)  the  velocity  at  the  end  of  7  sec- 
onds ;  (b)  the  distance  fallen  through ;  (c)  the  space  passed  over  in 
the  last  second. 

2.  A  body  is  projected  down  with  an  initial  velocity  of  20  feet 
per  second :  (a)  How  long  will  it  require  to  fall  594  feet  ?  (b)  What 
velocity  will  it  then  have  ? 


40  KINEMATICS.  [^6. 

3.  What  velocity  of  projection  must  a  stone  have  to  reach  the 
bottom  of  a  cliff  370  feet  high  in  3  seconds  ? 

4.  With  what  velocity  must  a  stone  be  thrown  down  the  shaft 
of  a  mine  556  feet  deep  in  order  that  the  sound  of  its  fall  may  be 
heard  at  the  top  after  4^  seconds  ?  The  velocity  of  sound  is  to  be 
taken  as  1112  feet  per  second. 

5.  A  body  projected  vertically  down  has  a  velocity  of  215  feet 
per  second  at  the  end  of  5  seconds :  Required  {a)  the  velocity  of 
projection ;  (b)  the  distance  gone  through. 

6.  A  body  projected  down  passes  over  133  feet  in  the  fourth 
second:  Required  the  velocity  of  projection. 

7.  A  stone  is  dropped  from  a  bucket  which  is  descending  a 
shaft  at  the  uniform  rate  of  12  feet  per  second,  and  at  the  moment 
when  the  bucket  is  238  feet  from  the  bottom :  {a)  How  far  will 
they  be  apart  in  2  seconds  ?  (p)  When  will  the  stone  reach  the 
bottom  ? 

8.  A  sand-bag  is  dropped  from  a  balloon  which  is  descending  at 
the  uniform  rate  of  24  feet  per  second ;  after  8  seconds  it  strikes 
the  ground :  {a)  What  was  the  height  of  the  balloon  ?  {b)  How  far 
were  they  apart  after  5  seconds  ? 

VII.  Bodies  projected  vertically  upward.    Article  44. 

1.  The  velocity  of  the  projection  upward  is  288  feet :  Required 
(a)  the  time  of  ascent;  (5)  of  descent;  (c)  the  height  of  ascent;  {d) 
the  distance  gone  in  the  first  and  last  seconds  of  ascent. 

2.  A  body  is  projected  up  with  a  velocity  of  192  feet  per 
second:  (a)  When  will  it  be  432  feet  above  the  starting-point  ?  {b) 
When  will  it  be  720  feet  below  the  starting-point  ?  Explain  the 
double  answer  in  each  case. 

3.  A  body  is  projected  up  with  a  velocity  of  208  feet :  How  long 
after  starting  will  its  velocity  be  {a)  -\-  64,  also  (5)  —  64  and  (c) 
—  272  ?    (The  minus  sign  indicates  downward  motion.) 

4.  What  velocity  of  projection  must  a  ball  have  in  order  to 
ascend  just  900  feet  ? 

5.  What  time  does  a  body  require  to  ascend  2304  feet,  that 
being  the  highest  point  reached  ? 

6.  What  velocity  of  projection  is  needed  to  make  a  body  ascend 
just  6  seconds  ? 


46.]  COMPOSITION   OF   MOTIONS.  41 

7.  A  ball  thrown  up  passes  a  staging  96  feet  from  the  ground  at 
the  end  of  1  second:  {a)  What  was  the  velocity  of  projection  ?  If 
the  time  is  6  seconds  {b),  what  is  the  answer  ? 

8.  A  body  projected  up  passes  over  112  feet  in  the  fifth  second  of 
its  ascent:  What  was  its  velocity  of  projection  ? 

9.  A  and  B  are  two  points  40  feet  apart  in  a  vertical  line ;  a  ball 
is  dropped  from  A,  and  at  the  same  instant  one  thrown  up  from 
B  with  a  velocity  =  80  feet  per  second :  When  and  where  will 
they  pass  each  other  ? 

10.  A  ball  is  dropped  from  the  top  of  a  cliff,  and  at  the  same 
instant  another  is  thrown  up  with  a  velocity  of  176  feet  per 
second :  («)  If  they  pass  each  other  at  the  end  of  2^  seconds,  how 
high  is  the  cliff  ?  (6)  How  far  were  they  apart  at  the  end  of  2 
seconds  ?  (c)  at  the  end  of  3  seconds  ? 

11.  A  ball  is  thrown  up  from  the  ground  with  a  velocity  of  128 
feet  per  second,  and  2  seconds  later  another  is  thrown  with  a 
velocity  of  160  feet:  When  and  where  will  they  pass  each  other  ? 

12.  A  bucket  is  ascending  a  shaft  uniformly  at  a  rate  of  82  feet 
per  second:  What  will  be  the  apparent  motion  of  a  stone  dropped 
from  it  («)  to  a  person  in  the  bucket  ?  (p)  to  a  person  on  the  side  of 
the  shaft  opposite  the  initial  point  ? 

13.  A  balloon  is  rising  uniformly  at  the  rate  of  96  feet  per 
second;  at  the  instant  it  is  640  feet  from  the  ground  a  sand-bag  is 
dropped  from  the  car:  What  will  be  the  motion  of  the  bag, 
when  will  it  reach  the  ground,  and  over  what  distance  will  it 
have  passed  ? 

VIII.  Prelected  up  or  down  a  smooth  Inclined  Plane.    Article  45. 

1.  The  height  of  the  plane  is  114  feet,  the  length  is  456,  the 
velocity  of  projection  down  is  25  feet  per  second :  {a)  How  long 
will  it  require  to  descend  ?    (Z>)  What  will  be  the  final  velocity  ? 

2.  The  height  and  length  are  144  and  576  feet  respectively:  {a) 
What  velocity  of  projection  up  is  required  that  it  may  just  reach 
the  top  ?    (^>)  What  time  will  it  take  ? 

3.  The  angle  of  the  plane  is  30°,  the  velocity  of  projection  down 
is  45  feet:  Required  (a)  the  velocity  at  the  end  of  4  seconds;  {Jj) 
the  distance  gone  through ;  (c)  the  distance  in  the  last  second. 

4.  The  angle  of  the  plane  is  30°,  the  velocity  of  projection  up  is 


42  KINEMATICS.  [46. 

80  feet  per  second:  Required  {a)  the  length  of  time  the  body  will 
continue  to  go  up,  {b)  the  distance  gone,  and  (c)  the  velocity  at  the 
end  of  3  and  of  8  seconds. 

IX.  Bodies  projected  against  Friction.    Articles  41,  42. 

[The  retardation  (or  minus  acceleration)  due  to  friction  takes  the 
place  of  the/ in  the  formulas  of  articles  42  and  44.] 

1.  A  body  projected  on  a  rough  horizontal  plane  has  at  starting 
a  velocity  of  120  feet  per  second,  but  loses  this  at  the  rate  of  12 
feet  for  each  succeeding  second:  (a)  What  is  the  retardation 
(minus  acceleration)  due  to  friction  ?  (b)  When  will  the  body 
stop  ?    (c)  How  far  will  it  have  gone  ? 

2.  The  retardation  due  to  friction  is  for  each  second  8  feet  pei 
second  for  a  given  sliding  body,  the  initial  velocity  is  40  feet  per 
second:  Required  {a)  the  time  it  will  continue  to  slide;  (b)  the 
distance  it  will  go  ;  (c)  its  velocity  at  the  end  of  3  seconds. 

3.  A  railroad-car,  when  the  engine  is  detached,  has  a  velocity 
of  15  miles  per  hour,  the  retardation  due  to  friction  is  1  foot-per- 
second  per  second :  How  far  and  how  long  will  the  car  continue 
to  move  ? 

4.  If  the  retardation  of  friction  is  4  f eet-per-second  per  second : 
(a)  What  initial  velocity  (in  miles  per  hour)  must  a  body  have  in 
order  to  slide  just  968  feet  ?  (b)  If  the  velocity  is  doubled,  how 
much  farther  will  it  go  ? 

5.  A  body  is  projected  up  a  rough  inclined  plane  at  an  inclina- 
tion of  30°,  the  retardation  of  friction  alone  is  4  feet-per-second 
per  second:  If  the  initial  velocity  is  400  feet  per  second,  how  far 
and  for  how  long  will  the  body  ascend  ? 

Composition  of  Uniform  and  Accelerated  Motion  not  in 
the  same  Straight  Line, 

46.  Composition  of  Uniform  and  Accelerated  Motion. 

If  a  body  tend  to  move  in  one  direction  with  uniform 
motion,  and  in  another  direction  with  accelerated  mo- 
tion, its  actual  path  is  not  a  straight  line,  but  a  curve. 
The  same  principle  involved  in  the  Parallelogram  of 


4Y.1 


PEOJEOTILES. 


43 


Velocities  (33)  makes  it  possible  to  determine  the  posi- 
tion of  the  body  at  the  end  of  any  given  time  (read  Art. 
68,  h,  p.  68).  For  if  one  side  of  the  parallelogram  repre- 
sents, in  direction  and  amount,  the  uniform  motion  in 
the  given  time,  and  the  adjacent  side  the  correspond- 
ing accelerated  motion,  the  diagonally  opposite  point  of 
the  parallelogram  will  indicate  the  actual  position  of  the 
body  at  the  end  of  this  time.  In  this  statement  nothing 
is  said  about  the  path  which  the  body  has  described. 

47.  Projectile.     The  simplest  application  of  the  above 
principle  is  to  the  case  of  the  projectile.     It  will  be 


Fig.  31. 

shown  that,  if  the  resistance  of  the  air  be  neglected,  the 
path  of  a  projectile  is  a  parabola. 

Suppose  a  body  starts  from  A  in  the  direction  AB 
(Fig.  21)  with  an  initial  velocity  u;  at  the  end  of  t 
seconds,  if  no  other  motion  were  imparted  to  itj  it  would 
reach  a  point  D,  so  that 

AD  =  w/.  (1) 


44 


KINEMATICS. 


[48. 


But  from  the  instant  of  starting  it  falls  vertically 
downward  under  the  influence  of  gravity,  with  uni- 
formly accelerated  motion.  At  the  end  of  t  seconds,  if 
it  had  no  initial  velocity,  it  would  fall  to  B,  so  that 

AB  =  \gt\  (3) 

But  as  shown  above  (46),  as  the  body  must  obey  both 
tendencies  to  motion  simultaneously,  its  actual  position 
at  the  given  time  will  be  at  C. 

Squaring  (1)  and  dividing  by  (2),  we  have 

AB  ~  AB   ~  ige  ~    g  ' 
or 

-    BC       2w» 

9  ' 


AB 


Therefore  the  ratio  of  the  square  of  the  ordinate  BO 
{B'C\  B"C")  to  the  abscissa  AB  {AB',  AB")  is  con- 
stant, and  hence  the  curve  is  a  parabola. 

48.  Position  of  the  Directrix,  Axis,  Focus.  The  line 
AD  (Fig.  22)  is  a  tangent  to  the  parabola  at  Ay  the  verti- 


FiG.  22. 

cal  line  EAB  is  a  diameter.     The  constant  value  of  the 

BG^  f       2?f''\ 
ratio  of  — r^  {  —  —  ]  is  four  times  the  distance  from  A 

AB  \         g  I 

to  the  directrix  or  to  the  focus.     The  same  relation  is  at 


49.]  PKOJECTILES.  45 

once  obvious  in  the  case  of  the  parabola,  whose  equation 
is  y^  =  4:ax  (i.e.  --  =  4fl^j;  it  may  also  be  proved  analyti- 
cally for  this  case,  where  the  co-ordinates  UB,  AD  are 
oblique.  It  is  proved  geometrically  in  a  following  para- 
graph (50,  e). 

If  AE  is  taken  on  the  vertical  line  equal  to  r-,  and 

EGE'  be  drawn  horizontally,  this  line  will  be  the  direc- 
trix; and  if  from  A  on  the  line  ^i^  (drawn  so  that  the 

angle  EAD  =  DAF)  we  take  AF  =  — ,  the  point  i^is 

the  focus  of  the  parabola.     The  vertical  line  GML  is 

the  axis. 

If  the  direction  of  the  initial  velocity  be  horizontal,  as 

in  Fig.  23,  then  the  starting-point 

A  is  the  vertex,  the  vertical  line  AB 

is  the  axis,  .and  the  focus  and  direc- 

u^ 
trix  are  at  distances  equal  to  —  from 


2? 


;?-» 


A,    This  figure  shows  well,  as  does 

also  Fig.  21,  what  is  meant  by  the 

statement  in  (27),  that  in  uniformly 

accelerated  motion  the  space  is  pro-  fig.  23. 

portional  to  the  square  of  the  time.     Here,  if  the  sue 

cessive  intervals  of  time  are  equal, 

AD  :  AD'  :  AD''  :  AD'",  etc.,  =1:2:3:4, 
and 

AB  :  AB'  :  AB"  :  AB'",  etc.,  =  1  :  4  :  9  :  16. 

49.  The  actual  path  of  a  projectile  deviates  widely 
from  a  parabola  because  of  the  resistance  of  the  air, 
which  is  very  great  with  high  velocities,  as  that  of  a 
cannon-ball  or  rifle-bullet  (perhaps  1600  feet  per  second 


46 


KINEMATICS. 


[50. 


in  starting).  For  this  reason  the  maximum  distance  is 
gcained,  not  by  an  angle  of  45°  (as  shown  below),  but  for 
an  angle  of  a  little  over  30°. 

A  jet  of  water  illustrates  the  subject  of  the  projectile 
well,  since  each  particle  may  be  considered  as  an  inde- 
pendent projectile,  and  thus  the  shape  of  the  jet  gives 
the  continuous  path.  It  shows,  moreover,  the  deviation 
caused  by  the  resistance  of  the  air. 

60.  Time  of  Flight,  Range,  etc.    In  Fig.  24  the  angle 


n         ^         ^'       X 
Fio.  24. 

HAKis  called  the  angle  of  projection,  and  the  horizon- 
tal distance  ^^is  the  range. 

(a)  From  the  triangle  AHK,  AH  =  ut  and  HK  = 
igf;  also, 

HK       igf        gt 

AH         tit        2u 

2u  sin  a 


\  t 


The  value  of  Hn  (1)  gives  the  time  of  fiiglit, 
(b)  Again, 

AK  =  AH  cos  a  =  tit. cos  a; 

or,  substituting  the  above  value  of  t, 


(1) 


60.] 


AK  = 


PEOJECTILES. 


2u^  sin  a  cos  a        u^  sin  2a 


47 


ff 


(^) 


This  value  AK  gives  the  range.  Further,  since  sin  2a 
=  sin  (180°  —  2a)  =  sin  2(90°  -  a),  it  is  obvious  that 
for  every  horizontal  distance  there  are  two  values  of  the 
angle  of  projection;  viz.,  a  and  (90°  —  a).  This  is  indi- 
cated in  Fig.  25.     The  time  of  flight  for  the  angle  a  is 

2u  sin  a       J  ,      ,.  .Q         ....    2u  cos  a 
,  and  for  (90   —  a)  it  is . 

if  if 

The  maximum  range  is  obtained  when  a  =  4:6°  and 


sin  2a  =  1,  for  the  value  of  ^^  is  then  the  greatest; 
for  this  case  the  two  paths  of  the  projectile  coincide. 

(c)  Since  the  vertical  component  of  the  initial  velocity 
is  u  sin  a,  the  actual  vertical  velocity  of  the  projectile 
will  be  given,  for  any  time  t,  by  the  formula  (41) 

V  =  u  sina  —  gt.                        (3) 
For  the  highest  point  F=  0,  and  hence  t  = ,  an(3 

ij 

combining  this  with  (1),  it  is  seen  that  the  times  of 
ascent  and  descent  are  the  same. 

{d)  The  distance,  GB  (Fig.  24),  of  the  projectile  above 


48 


KIIS^EMATICS. 


[50 


the  horizontal  line  ^^  is  given,  for  any  time  t,  as  fol- 
lows: 

CB  =  DB  -  DC  =  utmi a  -  igt\         (4) 


qt    glTl     /y 

For  the  highest  point  (M)  t  = ,  and  hence 

M]sr  = 


(5) 


g  ^  ^g 

This  value  is  greatest  when  sin  a  =  1  and  a  =  90°,  in 
which  case  the  parabola  becomes  a  double  straight  line. 
All  the  above  results  might  have  been  obtained  [as 
was  (3)  indeed]  by  the  formulas  in  articles  41,  42,  only 
taking  u  sin  a  for  u,  s  being  the  distance  above  or  below 
(—  s)  the  horizontal  plane. 


Fio.  26. 

(e)  We  may  prove  geometrically  the  point  mentioned 
in  Art.  48 ;  namely,  that  the  distance  from  A  to  the 

focus  is  equal  to  -^;  that  is,  to  one  fourth  of  the  con- 
stant value  of  the  ratio  of  the  square  of  the  ordinate  to 

the  abscissa  (  -^^  =  —  ) . 
\AB        g  J 

Draw  the  line  AF  (Fig.  26)  so  that  the  angle  DAF- 

DAE'y  then,  by  the  properties  of  the  parabola,  the  focus 


61.] 


PEOJECTILES. 


49 


must  lie  in  this  line;  it  must  also  lie  on  the  axis  GL, 
and  hence  will  be  at  F,  their  point  of  intersection.   Now 


AF: 


AL 


AL 


cos  FAL        cos  (90° -2a:) 
From  equation  (2)  above, 

u^  sin  2  a 


AL 

sin  2  a 


AL  =  iAK  = 


or 


AF 


AF 


u*  sin  2  a 


^9 


2g     ' 
4-  sin  2a, 


The  line  EGE'  is  the  common  directrix  of  all  the 
parabolas  described  by  projectiles  having  the  same  ini- 
tial velocity  but  different  angles  of  projection.  The  foci 
of  all  these  parabolas  lie  on  the  circumference  of  a  circle 

having  A  as  its  centre  and  a  radius  equal  to 


w 


51.  The  theory  of  the  projectile  may  be  further  illustrated  by 
the  case  of  a  jet  of  water  flowing  from  a  lateral  orifice  in  the  ver- 
tical side  of  a  reservoir  (Fig.  37).  By 
a  principle  of  hydrostatics  the  initial  K— 
velocity  of  flow  is  the  same  as  that 
which  would  be  gained  in  falling 
fi-eely  through  the  height  from  the 
top  of  water  to  the  orifice  (27) ;  that  is, 

Supposing  now  that  the  level  of 
the  water  is  kept  uniform,  the  direc- 
trix will  coincide  with  it,  that  is, 
AK ;  the  focus  will  be  at  F,  so  that 

CA  =  CF  —  -Kz>    Further,   it  may  be   readily  shown  that  the 


50  KINEMATICS.  [61, 

range  DE  is  the  same  for  any  two  points  taken,  as  Cand  C,  so  that 
AG  =  CD,  and  finally  that  the  maximum  distance  BHis  gained 
by  an  aperture  in  the  middle  at  B,  and  is  equal  to  AD{=:2AB), 


EXAMPLES. 

X.  Projectiles.    Articles  47-51. 

[The  resistance  of  the  air  is  left  out  of  account.] 

1.  The  initial  velocity  of  a  projectile  is  160  feet  per  second,  and 
the  angle  of  elevation  is  30°:  Required  (a)  the  time  of  flight;  (6) 
the  range;  (c)  the  highest  point  reached. 

2.  When  will  the  ball  in  example  1  be  96  feet  above  the  ground? 
Explain  the  double  answer. 

3.  The  initial  velocity  is  820  feet  per  second :  What  angle  of 
elevation  will  give  a  range  of  800  feet?  Show  that  there  are  two 
answers. 

4.  The  angle  of  elevation  is  15°:  What  initial  velocity  is  re- 
quired that  the  range  should  be  .4  miles  ? 

5  A  rifle-ball  is  shot  horizontally  from  the  top  of  a  tower  100 
feet  high,  and  with  an  initial  velocity  of  1200  feet  per  second: 
When  and  how  far  from  the  base  of  the  tower  will  it  strike  the 
horizontal  plane  below  ? 

6.  A  ball  is  thrown  horizontally  from  the  top  of  a  cliff  above 
the  sea;  it  strikes  the  water  in  5  seconds  and  at  a  horizontal  dis- 
tance of  a  mile:  What  was  (a)  the  initial  velocity,  and  (b)  what 
was  the  height  of  the  cliff  ? 

7.  If  (Fig.  27)  apertures  are  made  at  two  points  36  feet  from 
the  top  and  bottom  of  the  reservoir  respectively,  the  whole  height 
being  136  feet,  what  will  be  (a)  the  horizontal  distance  reached  by 
the  water  in  each  case,  and  what  (b)  the  initial  velocity  ? 

8.  A  stone  is  dropped  from  the  top  of  a  railroad-car,  16  feet 
above  the  ground,  and  when  it  is  moving  at  the  rate  of  45  miles 
per  hour.  What  will  be  its  apparent  motion  (a)  to  a  person  on  the 
train,  (b)  to  one  standing  by  the  track  ?  (c,  d)  When  and  where 
will  it  reach  the  ground  ? 

9.  At  what  angle  of  elevation  must  a  projectile  be  fired  in  order 
that  it  may  strike  an  object  2500  feet  distant  on  the  same  hori- 
zontal plane,  the  velocity  of  projection  being  400  feet  per  second  ? 


CHAPTER  II.— DYNAMICS. 

62.  The  preceding  chapter  was  devoted  to  the  sub- 
ject of  Kinematics,  or  the  discussion  of  the  motion  of 
bodies  without  reference  to  their  mass  or  to  the  force  or 
forces  which  cause  the  motion.  These  latter  subjects, 
included  under  Dynamics,  or  Kinetics,  are  considered 
in  the  present  chapter. 

Mass — Density —  Volume — Momentum, 

53.  Mass  or  Quantity  of  Matter.     The  mass  of  a  tody 

IS  the  quantity  of  matter  it  contains. 

The  relation  in  mass  or  quantity  of  matter,  of  differ- 
ent bodies  of  the  same  substance,  and  of  uniform 
density  (as  defined  in  66),  is  obviously  given  by  the 
ratio  of  their  volumes.  For  example,  the  mass  or 
quantity  of  matter  in  a  hundred  cubic  feet  of  iron  is 
ten  times  that  in  ten  cubic  feet.  For  bodies  of  uniform 
density  then:  the  mass  is  proportional  to  the  volume. 

In  general,  however,  for  bodies  of  different  substances 
it  is  possible  to  compare  their  masses  only  as  the  effect 
of  a  known  force  upon  them  is  observed.  Thus,  we 
judge  roughly  as  to  whether  a  barrel  is  empty  or  full, 
and,  in  the  latter  case,  as  to  the  nature  of  the  contents 
by  noting  the  degree  of  resistance  which  it  offers  to  a 
force  tending  to  move  it;  e.g.,  a  push  or  a  kick.  Simi- 
larly, if  a  ball  of  wood  and  another  of  the  same  size,  of 
lead,  attached  to  strings  of  equal  length,  be  whirled 


52  DYNAMICS.  [64. 

aronnd  at  the  same  rate,  the  pull  of  the  lead  upon  the 
centre  will  be  the  greater,  and  we  form  a  rough  estimate 
as  to  the  relation  of  mass  in  this  way.  Could  the  pull  at 
the  centre  be  exactly  measured  under  precisely  the  same 
conditions  in  each  case,  by  means  of  a  spring,  the  result 
would  give  the  true  relation  of  mass. 

Still,  again,  could  the  yelocities  given  by  the  same 
force  to  two  bodies  in  equal  times  be  exactly  determined, 
their  ratio  would  give  also  the  ratio  of  the  masses  of  the 
bodies.  No  one  of  these  methods  of  estimating  the 
mass  can  be  conveniently  employed  in  practice. 

64.  Mass  determined  by  Weight.  The  simplest  and 
at  the  same  time  most  accurate  method  of  comparing 
the  masses  of  two  bodies  is  by  their  weight,  for  the 
weight,  or  measure  of  the  earth's  attraction  upon  them 
determined  by  the  balance,  is,  as  proved  by  various  ex- 
periments, proportional  to  the  mass.  Take  two  bodies 
of  the  same  material,  as  two  lumps  of  lead:  if  the  weight 
of  the  first  is  twice  that  of  the  other,  then  it  is  easy  to 
see  that  its  mass,  or  the  quantity  of  matter  it  contains, 
is  also  twice  as  great.  But  this  is  true  in  general:  of 
two  lots  of  lead  and  cotton,  the  bulk  or  volume  of  the 
latter  may  be  much  greater  than  that  of  the  other,  but 
if  they  have  the  same  weight  they  have  also  the  same 
mass;  and  if  the  weight  of  the  lead  is  ten  times  that  of 
the  cotton,  its  mass  is  also  ten  times  greater. 

Mass  may  be  measured  then  by  weight,  and,  in  ordi- 
nary language,  the  latter  word,  expressed,  for  example, 
in  pounds,  is  used  as  standing  for  the  mass.  In  this 
sense,  the  uisriT  of  weight,  the  pound,  may  he  taken  as 
also  the  ukit  of  mass  (see  articles  71,  72). 

But  the  weight  is  also  used  as  a  measure  of  the  force 
of  gravity  and  of  other  forces  compared  with  it.    Hence 


66.]  MASS — DENSITY — VOLUME.  53 

it  must  be  carefully  noted  here  that  the  term  weight  is 
employed  with  two  distinct  meanings,  which  should  not 
be  confounded;  namely — 

(a)  As  a  measure  of  the  mass  or  quantity  of  matter  in 
a  given  body. 

(Z»)  As  a  measure  of  force  by  reference  to  the  force  of 
gravity. 

55.  Distinction  between  Mass  and  Weight.  Although 
the  weight  of  a  body  may  properly  stand  for  its  mass,  if 
their  true  relation  is  understood, the  two  terms  are  not 
identical.  The  mass  or  quantity  of  matter  of  a  lead 
ball  is  the  same  wherever  it  is  situated  on  the  earth's 
surface;  but  the  weight,  which  may  be  registered  on  a 
spring-balance,  is  slightly  greater  at  the  poles  than  at 
the  equator.  Again,  at  the  surface  of  the  sun  the  force 
of  attraction  on  the  same  piece  of  lead,  registered  as 
before  by  the  stretching  of  a  spring,  would  be  about 
twenty-eight  times  greater  than  on  the  earth.  Still 
further,  if  we  conceive  of  it  as  at  a  point  in  space  far 
away  from  attracting  bodies,  there  would  be  no  sensible 
pull  on  the  spring,  nothing  to  correspond  to  the  terrestrial 
weight,  but  the  mass  would  be  everywhere  the  same. 

56.  Relation  between  Mass,  Density,  and  Volume. 
The  density  of  a  body  is  the  mass  or  quantity  of  matter  in 
the  unit  of  volume.  In  comparing  different  bodies  the 
density  of  water  at  the  temperature  of  39.2°  F.  (4°  C.)  is 
generally  taken  as  unity ;  the  fact  that  the  weight — that 
is,  the  mass — of  a  given  volume  of  lead  is  11-J-  times,  or 
of  iron  7  times,  that  of  the  same  volume  of  water  is 
expressed  by  saying  that  the  density  of  lead  is  ll^,  and 
of  iron  is  7.  A  body  is  said  to  be  throughout  of  uni- 
form density  when  equal  volumes,  however  small,  have 
the  same  mass. 


54  DYNAMICS.  fST. 

In  Art.  53  it  was  stated  that  for  bodies  of  uniform 
density  the  mass  is  proportional  to  the  volume.  It  also 
follows  that  for  bodies  of  equal  volume  the  mass  is  pro- 
2Jortional  to  the  density. 

Therefore,  in  general,  the  mass  (if)  is  proportional 
to  the  product  of  the  volume  (F)  and  density  {D). 
This  may  be  expressed  mathematically  in  this  form: 

M  ex  DV;    that  is,    ■^,  =  jyyn 
whence 

D  oc  -^y       and         V  a  jy. 

♦  67.  Momentum.  The  momeyitum  of  a  hody  is  equal  to 
the  product  of  the  mass  and  velocity.  Two  bodies  of 
the  same  mass,  and  moving  with  the  same  velocity,  have 
obviously  the  same  momentum.  If  these  two  bodies 
were  joined  together,  still  retaining  the  same  velocity  as 
before,  the  momentum  of  the  two  together  as  a  whole 
would  be  twice  that  of  either  of  them  separately.  In 
general,  of  two  bodies  having  the  same  velocity,  if  the 
mass  of  one  is  five  times  that  of  the  other,  its  momentum 
will  be  also  five  times  as  great;  or. 

If  the  velocity  is  constant,  the  momentum  is  propor- 
tional to  the  mass. 

Again,  suppose  two  bodies  of  the  same  mass,  but  one 
moving  with  twice  the  velocity  of  the  other,  -its  momen- 
tum will  be  also  twice  as  great;  or,  in  general. 

If  the  mass  is  constant,  the  momentum  is  proportio7ial 
to  the  velocity. 

The  UNIT  OF  MOMENTUM  is  the  momentum  of  a  body 
of  unit  mass,  moving  with  the  unit  velocity  of  one  foot  per 
second.     A  body  whose  mass  is  M  and  whose  velocity  is 


FORCE  DEFINED.  55 

V  has  a  momentum  equal  to  the  product  of  the  mass 
into  the  velocity,  or 

Momentum  ■=^  Mv, 


EXAMPLES. 

XI.  Mass— Density—  Volume.    Article  56. 

1.  The  masses  of  two  bodies  are  as  2  to  7,  and  tlieir  densities  as 
6  to  5:  What  is  tlie  ratio  of  their  volumes  ? 

2.  Tlie  masses  of  two  bodies  are  as  5  to  6,  their  volumes  as  2  to 
^3:  What  is  the  ratio  of  their  densities  ? 

3.  Two  bodies  of  the  same  mass  have  densities  as  8  to  9:  What 
is  their  ratio  in  volume  ? 

4.  What  is  the  ratio  in  volume  of  a  piece  of  silver  weighing 
20  lbs.  and  having  a  density  of  10.5  (referred^  to  water  as  unity), 
and  a  piece  of  iron  weighing  5  lbs.  and  having  a  density  of  7  ? 

5.  If  a  cubic  foot  of  water  weighs  62.5  lbs.  (density  unity),  what 
is  the  weight  of  a  cubic  inch  of  mercury,  density  13.6  ? 

6.  What  is  the  ratio  in  weight  (that  is,  in  mass)  of  two  blocks  of 
stone,  one  having  a  volume  of  50  cubic  feet  and  a  density  of  3,  the 
other  a  volume  of  45  cubic  feet  and  a  density  of  2.75  ? 

7.  If  a  liter  (1000  cubic  centimeters)  of  water  weighs  a  kilogram 
(density  unity,  temperature  4°  C.  =  39.2°  F.),  and  a  cubic  centi- 
meter of  another  liquid  weighs  1.1  grams,  at  the  same  tempera- 
ture, what  is  the  density  of  the  latter  liquid  ? 

8.  If  a  cubic  foot  of  fresh  water  weighs  62.5  lbs.,  and  of  salt 
water  64  lbs. ,  what  is  the  density  of  the  salt  water  ? 

Kinds  of  Forces — Force  of  Gravity, 

68.  Definition  of  Force.  A  force  is  that  ivhich  moves 
or  tends  to  move  a  body,  or  ivhicli  changes  or  tends  to 
change  its  motion,  either  in  direction  or  quantity. 

In  view  of  the  fact,  before  explained  (11),  that  all 
bodies  of  which  we  have  any  knowledge  are  in  motion, 
the  completeness  of  the  definition  would  not  be  im- 


56  DYNAMICS.  [69. 

paired  by  the  omission  of  the  first  clause.  It  is,  how- 
ever, convenient  to  consider  the  earth  and  all  bodies 
which  do  not  change  their  position  with  reference  to  it 
as  at  rest,  and  the  definition  conforms  to  that  idea. 
Further,  the  word  *Hend"  is  added  because  the  action 
of  one  force  may  be  neutralized  by  that  of  one  or  more 
opposing  forces,  so  that  the  motion  which  it  tends  to 
produce  is  not  observed.  For  example,  a  book  resting 
on  a  table  tends  to  fall  to  the  ground  under  the  action 
of  the  force  of  gravity,  but  an  equal  opposite  force,  the 
resistance  or  reaction  of  the  table,  keeps  it  at  rest. 

59.  Continued  and  Impulsive  forces.  A  force  is  said 
to  be  continued  when  its  action  continues  an  appreciable 
length  of  time.  It  is  uniformly  continued,  or  con- 
stant, when  its  intensity  is  always  the  same;  this  is  true 
of  the  force  of  gravity  at  a  given  point  on  the  earth's 
surface.  A  continued  force  is  variable  when  its  in- 
tensity is  diSerent  at  different  times,  as  the  force  of  a 
watch-spring,  whose  intensity  diminishes  as  the  spring 
Unwinds. 

An  impulsive  force  is  one  which  acts  through  so  short 
a  time  that  the  law  of  its  action  cannot  be  determined, 
and  we  are  limited  to  considering  its  effects  after  its 
•action  has  ceased.  This  is  true  of  the  blow  from  a  bat 
on  a  ball.  Such  a  force,  however,  is  not  strictly  in^ 
stantaneous,  but  one  whose  intensity  is  very  great  and 
varies  during  the  brief  time  of  its  action.  There  is  con- 
sequently no  essential  difference  between  the  two  classes 
of  forces. 

60.  Effects  of  Force  npon  a  Free  Body.  A  continued 
force  tends  to  produce  accelerated  motion  in  the  body 
acted  upon.     If  the  force  is  uniform  as  well,  it  tends  to 


61.1  FOECE  DEFINED.  67 

give  the  body  uniformly  accelerated  motion.  This  is 
practically  the  motion  of  a  body  falling  toward  the  earth 
under  the  influence  of  the  nearly  constant  force  of 
gravity  (but  see  Art.  64).  Therefore,  if  the  accelera- 
tion produced  by  a  constant  force  is  /  (for  gravity  ^), 
the  relations  between  the  velocity  acquired  (v)  and 
space  passed  over  (s)  in  a  given  time  (^)  are  expressed 
by  the  familiar  equations  (from  Art.  27): 

V  =  ft,         For  gravity,  v  =  gt, 
s  =  ift\  s  =  igt% 

v^  =  2fs.  v'  =  2gs, 

The  motion  of  a  body  acted  upon  by  an  impulsive 
force  tends,  after  this  force  has  ceased  acting,  to  be 
uniform,  as  a  ball  struck  along  the  ground  by  a  bat. 
This  is  true,  indeed,  of  any  body  in  motion,  and,  as 
explained  in  Art.  67,  is  a  consequence  of  the  first  law 
of  motion. 

The  presence  of  other  opposing  forces  may  modify  the 
effect  of  the  force  considered.  For  example,  a  stone 
thrown  vertically  upward,  and  which  consequently  tends 
to  move  uniformly  in  that  direction,  has  in  fact  re- 
tarded motion  because  of  the  continued  and  simul- 
taneous action  of  the  force  of  gravity  downward.  So, 
too,  a  ball  rolled  along  the  ground,  as  a  matter  of  ex- 
perience, soon  comes  to  a  state  of  rest  because  of  the 
opposing  force  of  friction. 

61.  Equilibrium.  A  body  is  said  to  be  in  equilibrium, 
with  respect  to  two  or  more  forces,  when  they  neutralize 
each  other  so  that  its  condition  of  rest  or  motion  is  not 
affected  by  them.  The  book  mentioned  in  Art.  58  is 
an  example  of  equilibrium.     But  equilibrium  does  not 


58  DYNAMICS.  [62. 

necessarily  imply  the  rest  of  the  body  in  question.  For 
example,  a  ball  rolling  on  a  perfectly  smooth  horizontal 
surface  is  in  equilibrium  with  respect  to  the  two  equal 
and  opposite  forces — the  action  of  gravity  and  the  re- 
action of  the  surface.  The  same  would  hold  true  how- 
ever many  forces  were  involved  if  they,  taken  together, 
did  not  affect  the  motion  of  the  body.  Equilibrium 
strictly  implies  simply  absence  of  acceleration. 

62.  Examples  of  Forces.  The  first  and  simplest  con- 
ception of  force  we  derive  from  muscular  exertion,  as 
we  note  its  effects  in  different  ways.  AYith  it  we  join 
all  other  agencies  which  produce  similar  effects  in 
changing  the  motion  of  bodies,  as  gravity,  cohesion, 
electrical  attraction  and  repulsion,  and  so  on. 

63.  Force  of  Gravity.  The  force  of  gravity  is  mani- 
fested in  the  attraction  which  the  earth  exerts  on  a  mass 
of  matter  near  its  surface,  and  which  causes  it  to  fall,  or 
«;end  to  fall,  toward  it.  This  is  a  special  case  of  the 
universal  law  of  gravitation,  established  by  Newton,  and 
.'According  to  which 

Every  particle  of  matter  attracts  and  is  attracted  iy 
every  other  particle  with  a  force  which  varies  directly  as 
the  product  of  the  masses  and  inversely  as  the  square  of 

MM' 

the  distance,     F  a  — ^^ — . 
d 

A  stone,  therefore,  as  truly  attracts  the  earth  as  it  is 

attracted  by  it;  so,  also,  the  earth  attracts  and  is  attracted 

by  the  moon,  the  sun,  and  the  other  bodies  of  the  solar 

system.     In  terrestrial  mechanics,  however,  we  have  tc 

do  simply  with  the  attraction  of  the  earth  upon  bodies 

on  or  near  it,  and  as  its  mass  is  indefinitely  great  in 

comparison,  the  reciprocal    attraction  is  left    out    of 


64.]  FOECE  OF  GEAVITY.  59 

account.  Therefore,  since  the  mass  of  the  earth  is  con- 
stant, the  force  of  its  attraction  on  any  body  varies 
directly  as  the  mass  of  that  body  and  inversely  as  the 
square  of  its  distance  from  the  earth's  centre ;  that  is, 

M 
F  a  -Tj.     From  this,  it  follows  that  for  a  given  body 

the  force  of  attraction  varies  inversely  as  the  square  of 
the  distance ;  and  for  different  bodies  at  the  same  dis- 
tance the  force  is  directly  as  their  masses.  These  two 
points  are  expanded  in  articles  64  and  65. 

The  attraction  between  different  bodies,  although  a  force  of 
small  intensity  where  their  masses  are  small,  may  be  demon- 
strated in  other  ways  than  by  the  fall  of  a  body  to  the  earth. 
It  is  illustrated  by  the  deviation  from  the  usual  perpendicular 
position,  which  is  observed  when  a  ball  hung  by  a  string  is 
suspended  near  an  isolated  mountain.  Experimenting  in  this 
way,  Dr.  Maskelyne  found  the  angle  of  deviation  for  two  plumb- 
lines,  placed  on  opposite  sides,  north  and  south,  of  Mt.  Schehal- 
lien  in  Scotland,  and  at  a  distance  of  4000  feet  from  each  other, 
to  be  13  seconds.  From  this  result  the  mean  density  of  the  earth 
was  calculated  to  be  about  5  times  that  of  water. 

This  attraction  has  also  been  shown  by  Cavendish  in  a  more 
delicate  manner,  by  means  of  the  torsion  balance.  Two  small 
balls  of  lead  were  attached  to  the  ends  of  a  slender  wooden  rod 
which  was  supported  at  tlie  centre  by  a  fine  wire  of  considerable 
length.  Two  larger  balls  were  then  approached  to  the  small  ones 
and  on  opposite  sides,  so  that  their  effect  was  felt  in  the  same 
direction.  The  result  was  that  the  small  balls  were  attracted  by 
the  larger  ones,  and  the  rod  supporting  them  deflected  from  its 
original  position  of  rest.  The  angle  of  deflection  showed  the 
amount  of  the  torsion  (or  twist)  of  the  wire,  and  this  measured  the 
intensity  of  the  attracting  forces.  By  comparing  the  attractions 
of  these  balls  with  that  of  the  earth,  Cavendish  calculated  the 
mean  density  of  the  earth  to  be  5.45. 

64.  (1)  Tfie  force  of  attraction  of  the  earth  on  a  given 
lody  varies  inversely  with  the  square  of  the  distance. 


60  DYNAMICS.  fS4. 

F  cc  ■^,     This  law  means  that  if  the  distance  from  the 
a 

attracting  body  be  increased  two,  three,  or  ten  times,  the 

force  of  attraction  it  exerts  on  another  body  is  i,  -J^,  or 

y^  respectively;  if  the  distance  is  diminished  to  -J,  ^, 

and  so  on,  the  force  of  attraction  becomes  4  times,  9 

times,  etc.,  greater. 

Two  consequences,  of  importance  here,  follow  from 
this  principle: 

{a)  The  attraction  of  the  earth  is  sensibly  constant  at 
a  given  locality  and  for  the  different  heights  above  the 
surface  involved  in  ordinary  observations. 

This  attraction  may  be  proved  to  be  exerted  as  if  the 
whole  mass  were  concentrated  at  a  point  at  or  near  the 
centre,  called  the  centre  of  gravity.  If  now  we  call  the 
radius  of  the  earth  in  round  numbers  4000  miles,  the  dis- 
tances from  this  centre  for  two  bodies — the  one  at  the  sea- 
level  and  the  other  a  mile  above — will  be  4000  and  4001 
miles  respectively,  and  the  ratio  of  the  forces  of  attraction 
will  be,  as  above,  400  l'^  :  4000  ;  but  this  difference  is  so 
small  that  it  may  often  be  left  out  of  account.  In 
accurate  physical  Investigations,  however,  this  difference 
can  by  no  means  be  neglected.  Indeed,  it  should  be 
most  carefully  noted  that  while  gravity  is  here  said  to 
be  sensibly  constant,  it  really  varies  uninterruptedly  as 
the  distance  from  the  centre  increases,  and  is  not  abso- 
lutely tlie  same  for  two  points,  one  of  which  is  a  foot 
above  the  other. 

If  we  imagine  a  body  to  pass  from  the  surface  toward 
the  centre  of  the  earth,  it  may  be  demonstrated  that  the 
attraction  will  diminish,  and  if  the  earth  were  homo- 
geneous, in  the  same  ratio  as  the  distance  from  the  cen- 
tre diminishes;  at  the  centre  the  attraction  is  zero. 


65.]  FORCE  OF  GEAVITY.  61 

(b)  The  force  of  attraction  is  least  at  the  equator  and 
increases  toward  the  poles. 

The  earth  haying  the  shape  of  an  oblate  spheroid,  the 
polar  diameter  is  about  26  miles  shorter  than  the  equa- 
torial, and  hence,  as  it  may  be  demonstrated,  the  force 
of  attraction  is  greater  for  a  body  at  the  poles  by  j\j. 

To  this  cause  for  the  variation  of  the  intensity  of  the 
force  of  gravity  is  to  be  added  a  second,  the  rapid  rota- 
tion of  the  earth  on  its  axis.  The  effect  of  this  cause  to 
diminish  the  gravitation  is  greatest  at  the  equator,  and 
grows  less  as  we  go  from  it,  and  becomes  zero  at  the 
poles  (80).  On  this  account,  then,  the  attraction  is 
greater  by  ^^  at  the  poles  than  at  the  equator.  As  the 
result  of  the  two  causes  taken  together,  the  force  of 
gravity  at  the  poles  is  about  y^  greater  than  at  the 
equator.  Thus  the  weight  of  a  given  mass  of  matter, 
by  which  the  pull  of  the  earth  may  be  measured,  in- 
creases as  we  go  from  the  equator  northward  or  south- 
ward toward  the  poles.  This  difference  could  be  noted, 
for  example,  by  observations  with  a  •sufficiently  delicate 
spring-balance,  and  would  amount  to  about  1  lb.  in 
200  lbs.  This  difference  is  left  out  of  account  in  ordi- 
nary commercial  transactions,  but  cannot  be  neglected 
in  physical  problems  where  accuracy  is  required. 

The  force  of  gravity  also  varies  somewhat  for  differ- 
ent points  on  the  earth's  surface  in  consequence  of  varia- 
tions in  the  density  of  the  material  of  the  earth.  This 
subject  is  expanded  in  a  later  article  (250),  where  the 
values  of  g  for  different  points  are  also  given. 

65.  Again:  (2)  The  attraction  of  the  earth  on  bodies 
at  the  same  distance  is  proportional  to  their  masses. 
Foe  M.  Hence  the  acceleration  given  to  a  falling  body 
by  the  earth's  attraction  is  independent  of  its  mass. 


62  DYNAMICS.  [65. 

For  example,  two  pieces  of  lead  of  widely  differing 
weights,  or  a  piece  of  lead  and  a  feather,  fall  the  same 
distance  toward  the  earth  in  the  same  time  and  gain  the 
same  velocity  (that  is,  32  feet  per  second  for  each  second). 
But  the  experiment  succeeds  only  when  all  disturbing 
causes — e.^.,  the  resistance  of  the  air — are  removed,  so 
that  an  exhausted  receiver  must  be  employed.  As  a 
matter  of  experience,  in  comparing  the  fall  of  a  small 
and  heavy  body  and  of  a  larger  and  lighter  one,  the 
former  will  fall  the  fagter  under  ordinary  conditions, 
but  this  is  only  because  it  feels  less  the  resistance  of 
the  air. 

The  fact  here  mentioned  is  based  upon  the  above 
law,  and  is  what  a  simple  consideration  would  lead 
us  to  expect.  Suppose  several  small 
•  •  •  •  •  i    shot  as  those  at  a  (Fig.  28);    it  is 

Fig.  28.  obvious  that  under  the  earth's  at- 

traction they  would  fall  together,  keeping  their  relative 
position  with  reference  to  each  other,  and  reaching  the 
ground  at  the  same  time  and  with  the  same  velocity. 
If  now  we  suppose  them  rigidly  connected,  but  their 
positions  unchanged,  they  would  form  a  mass  5  times 
greater  than  a  single  one  (as  J),  and  attracted  by  a  force 
also  5  times  greater;  they  would  still  fall  together,  and 
the  acceleration  of  this  mass  and  the  single  one  would 
be  the  same.  This  course  of  reasoning  might  be  ex- 
tended to  any  two  bodies,  however  unlike  in  mass.  The 
force  of  attraction  increases  always  in  the  same  ratio 
that  the  number  of  particles,  or  the  mass,  of  the  body 
attracted  increases,  and  hence  the  effect  of  the  force  of 
attraction  must  remain  constant. 


0\- 


66.]  LAWS  OF  MOTION.  bd 

EXAMPLES. 
XII.  F(yrce  of  Qramty,     Articles  63,  64,  65. 

1.  At  what  distance  from  the  centre  of  the  earth  would  a  mass 
of  matter  weighing  32  lbs.  on  the  earth's  surface  exert  a  pull 
equivalent  to  1  lb.  on  a  spring-balance  ? 

2.  If  the  mass  of  the  sun  is  350,000  times  that  of  the  earth,  and 
its  diameter  112  times,  what  is  the  acceleration  of  gravity  at  its 
surface  ?  (see  also  p.  67) 

3.  If  the  moon's  mass  is  /^  of  that  of  the  earth,  and  its  diameter 
2160  miles,  that  of  the  earth  being  about  7900  miles,  what  is  the 
acceleration  of  gravity  on  the  moon's  surface  ? 

4.  If  the  acceleration  of  gravity  on  the  surface  of  Jupiter  is 
2.62  times  ^,  and  its  diameter  11  times  that  of  the  earth,  what  is 
their  ratio  in  mass  ? 

5.  If  the  value  of  g  at  the  equator,  at  the  sea-level,  is  32.096, 
what  is  its  value  at  the  summit  of  a  mountain  in  latitude  0°,  at  an 
altitude  of  15,840  feet  ?    The  equatorial  radius  is  3963.3  miles. 

Newton's  Laws  of  Motion, 

66.  Laws  of  Motion.  The  three  laws  of  motion,  as 
stated  by  Newton,  are: 

(1)  Every  body  continues  in  a  state  of  rest,  or  of  uni- 
form motion  in  a  straight  line,  except  in  so  far  as  it  may 
be  compelled  by  impressed  forces  to  change  that  state. 

(2)  Change  of  motion  is  proportional  to  the  impressed 
force,  and  takes  place  in  the  direction  of  the  straight 
line  in  which  the  force  acts. 

(3)  To  every  action  there  is  always  an  equal  and  con- 
trary reaction;  or,  the  mutual  actions  of  two  bodies  are 
always  equal  and  opposite  in  direction. 

The  truth  of  these  laws  is  established  by  observation 
and  experiment;  it  is  found  that  all  legitimate  conclu- 


64  DYNAMICS.  [67. 

sions  deduced  from  them  are  in  harmony  with  the  ob- 
served facts  of  nature. 

67.  The  FIRST  law  asserts  what  is  sometimes  called 
the  inertia  of  matter;  in  other  words,  that  matter  alone 
is  powerless  to  change  its  state  either  of  rest  or  motion. 
These  points  may  be  considered  separately. 

{a)  The  tendency  of  a  body  at  rest  to  remain  in  this 
condition  is  universally  recognized;  it  is  always  mani- 
fested by  the  apparent  resistance  which  such  a  body 
offers  to  a  force  tending  to  set  it  in  motion.  This 
apparent  resistance  of  heavy  bodies  has  nothing  to  do 
with  that  caused  by  other  opposing  forces;  e.g.,  friction. 
It  increases,  as  stated  in  53,  with  the  mass  of  a  body, 
and,  as  explained  under  the  second  law  of  motion  (68),  is 
due  solely  to  the  fact  that  in  such  cases  a  force  must 
continue  to  act  through  a  certain  time  in  order  to  im- 
part sensible  motion.  For  example,  suppose  a  heavy 
fly-wheel  of  an  engine  free  to  turn  on  its  axis  without 
friction,  or  a  heavy  cannon-ball  suspended  by  a  string 
of  great  length,  or  a  massive  iron  door  well  poised  on 
its  hinges;  in  all  such  cases  the  hand  in  trying  to  move 
the  bodies  seems  to  encounter  resistance,  which  ex- 
presses the  inertia  of  the  body,  or  its  tendency  to 
remain  at  rest.  The  only  real  difference,  however, 
between  the  bodies  named  and  a  light  body  moved  with 
a  touch  arises  from  the  greater  mass  involved  in  the 
former  case;  both  alike  have  inertia. 

This  inertia,  or  resistance  to  motion,  is  a  property  of 
all  forms  of  matter ;  it  is  manifested  by  the  water  when 
a  boat  moves  rapidly  through  it,  and  by  the  air  when 
one  is  driving  or  running  rapidly. 

{l)  The  application  of  the  second  part  of  the  law  is 
equally  familiar,  although  it  is  impossible  to  give  an 


67.]  LAWS  OF  MOTIOIT.  65 

experimental  proof  of  its  truth.  It  requires  that  a 
body  once  in  motion  shall — unless  acted  upon  by  some 
force  tending  to  change  its  motion — continue  to  move 
f oreyer  uniformly  and  in  a  straight  line.  This  tendency 
toward  continuance  in  a  state  of  motion  is  seen  in  the 
apparent  resistance  that  bodies  in  motion  make  to  a 
force  tending  to  stop  them.  We  observe,  also,  that  if  a 
carriage  in  rapid  motion  is  suddenly  stopped,  the  per- 
sons occupying  it  are  thrown  violently  forward;  so,  too, 
if  a  person  steps  off  from  a  train  in  rapid  motion,  his 
body  tends  to  keep  its  forward  motion,  and  when  the 
motion  of  the  feet  is  arrested  by  touching  the  ground, 
the  rest  of  the  body  is  thrown  forward  and  a  fall  is  the 
result, 

But  a  body  in  niotion  not  only  tends  to  move  on  uni- 
formly, but  also  in  a  straight  line.  Hence  if  we  observe 
a  body  moving  in  a  circular  path,  as  a  stone  attached  to 
a  string  and  whirled  about  a  centre,  we  are  justified  in 
concluding  that  a  force  is  continually  acting  to  deflect  it 
from  a  straight  line.  If  the  string  to  which  the  stone  is 
attached  breaks,  the  stone  flies  off  at  a  tangent  to  its 
former  course.  So,  too,  of  the  earth:  it  moves  about 
the  sun  in  a  path  nearly  circular,  its  tendency  to  move 
in  a  straight  line  being  overcome  by  the  continued 
attraction  of  the  sun. 

The  tendency  to  uniform  motion  in  a  right  line  is  all 
that  can  be  observed;  for,  as  stated  above  (60),  uniform 
motion  in  a  body  not  acted  upon  by  any  force  is  alto- 
gether contrary  to  experience — any  body,  once  set  in 
motion,  sooner  or  later  comes  to  rest.  But  in  every  such 
case  it  is  possible  to  trace  the  more  or  less  rapid  loss  of 
motion  to  outside  causes;  the  most  universally  present 
are  friction,  or  the  resistance  to  motion  due  to  the  rough- 


(j6  dynamics.  [68. 

nesses  of  the  surfaces  in  contact,  and  the  resistance'  of 
the  air,  which  last  is  an  important  element  in  the  case 
of  rapid  motion,  as  that  of  a  bullet. 

The  truth  of  the  law  is  argued  from  the  observed  fact 
that,  in  proportion  as  these  opposing  forces  are  removed, 
the  motion  continues  longer  and  longer.  A  ball,  which 
with  a  given  impulse  will  roll  a  certain  distance  on  a 
horizontal  surface  of  turf,  rolls  farther  on  gravel,  farther 
still  on  a  marble  floor,  and  still  farther  on  a  sheet  of  ice. 
So,  too,  the  time  which  a  pendulum,  once  set  in  motion, 
will  continue  to  vibrate  without  additional  impulse 
becomes  longer  and  longer  as  we  remove  the  friction  on 
its  axis  of  support,  and  the  resistance  of  the  air  by 
placing  it  in  an  exhausted  receiver.  The  case  of  the 
earth  is  the  most  perfect  illustration  of  this  law,  for  it 
moves  on  in  its  orbit  with  a  mean  velocity  that  the  most 
accurate  observations  can  hardly  prove  to  vary,  although 
it  is  receiving  no  forward  impulse;  the  sun's  attraction, 
as  explained  above,  only  serves  to  keep  it  in  its  nearly 
circular  orbit. 

68.  The  SECOiq"D  law  of  motion  asserts  (a)  that  the 
change  of  motion  is  proportional  to  the  impressed  force. 

By  motion  is  meant  here  momentum,  or  the  pro- 
duct of  the  mass  and  velocity,  as  defined  in  Art.  57. 
The  law  consequently  asserts  that  the  change  of  mo- 
mentum in  a  given  time  is  proportional  to  the  force 
which  acts,  and  hence,  as  explained  in  71,  this  change 
of  momentum  is  a  measure  of  the  force. 

Let  F  represent  the  force,  31  the  mass  of  the  body 
moved,  and  /  the  velocity  given  to  it  in  one  second 
(the  acceleration).  Then  Mf  is  the  momentum  gener- 
ated in  one  second,  and  by  this  law  F  is  proportional  to 
Mf ;  that  is,  F  is  equal  to  Mf  multiplied  by  some  con- 


68.]  LAWS   OF  MOTION.  67 

stant  number;  if  suitable  units  are  taken,  this  constant 
becomes  unity,  and  then 

F  =  Mf.  (1) 

If  the  force  acts  for  t  seconds  with  uniform  intensity, 
then  its  effect  will  be  proportional  to  the  time  (=  Ft), 
and  the  velocity  given  to  the  body  at  the  end  of  this 
time  will  be  ft  or  v.  Hence,  with  the  same  provision 
as  above,  we  have 

Ft  =  Mv,  (2) 

In  the  case  of  gravity  the  momentum  generated  in  a 
given  time — that  is,  in  one  second — is  Mg,  But  since 
the  weight  of  the  body,  expressed  in  standard  pounds,  is 
proportional  to  the  mass  (M),  and  also  to  the  accele- 
ration of  gravity  (g),  the  weight  is  also  proportional  to 
the  product  Mg;  if  a  suitable  unit  of  mass  is  taken,  we 
may  write 

W=:Mg,  (3) 

or 

W 
M  =  —,  (4) 

g  / 

The  expression  in  equation  (4)  is  the  value  of  the  mass 
in  terms  of  the  weight  in  pounds  which  is  ordinarily 
employed  in  Mechanics. 

From  equation  (2)  the  following  principles  are  de- 
duced: 

1.  T7ie  velocities  given  in  the  same  time  to  different 
todies  of  the  same  mass  are  proportional  to  the  acting 
forces.  That  is,  if  forces  whose  intensities  are  as  1,  2,  3, 
act  on  three  bodies  of  equal  mass,  the  velocities  gene- 
rated in  the  same  time  will  be  in  the  ratio  of  1:2:3. 
For  example,  the  intensity  of  the  sun's  attraction  at  ijs 
surface  is  about  28  times,  and  of  Jupiter  2.6  times,  that 


68  DYNAMICS.  [68. 

of  the  earth;  therefore  the  Telocity  acquired  at  the  end 
of  one  second  by  a  falling  body,  on  the  sun,  on  Jupiter, 
and  on  the  earth,  will  be  respectively  28  X  32  feet  per 
second,  2. 6  X  32,  and  32. 

2.  If  equal  forces  act  upon  bodies  of  different  mass 
for  the  saine  time,  the  velocities  will  he  inversely  propor- 
tional to  the  masses.  A  force  which  would  give  a  body 
of  mass  1  a  velocity  of  12  feet  per  second  in  a  certain 
time,  would  give  a  body  of  mass  3  a  velocity  of  only 
4  feet  per  second  in  the  same  time.  This  principle 
shows  the  reason  why  a  "heavy"  body — that  is,  one  of 
great  mass — seems  to  offer  more  resistance  to  motion 
than  a  lighter  one.  For  suppose  the  ratio  in  mass  to 
be  10  :  1;  then  the  same  force  acting  upon  the  heavier 
body  will  give  it  in  the  same  time  the  same  momentum, 
but  only  yV  ^tie  velocity.  In  other  words,  it  must  act 
through  ten  times  the  length  of  time  in  order  to  gene- 
rate the  same  velocity. 

Again — 3.  To  give  bodies  of  different  mass  the  same 
velocity  in  the  same  time,  the  forces  must  in  each  case  he 
proportional  to  the  masses.  As  explained  in  65,  this  is 
true  of  the  force  of  the  earth's  attraction,  which  gives  to 
all  falling  bodies  at  its  surface  the  same  acceleration. 

Finally — 4.  If  equal  forces  act  upon  bodies  of  equal 
mass,  the  velocities  generated  will  he  proportional  to  the 
times  of  action. 

(b)  The  second  law  also  states  that  the  direction  of 
motion  is  that  of  the  impressed  force.  When  a  force 
acts  upon  a  material  particle  at  rest — that  is,  a  portion 
of  matter  so  small  that  its  dimensions  may  be  left  out  of 
account — the  truth  of  this  law  is  evident.  If  a  body  is 
acted  upon,  it  will  in  general  take  the  direction  of  the 
impressed  force  only  when  the  line  of  its  action  passes 


69.]  LAWS   OF   MOTIOK.  69 

through  a  point  in  it  called  the  centre  of  inertia.  When 
this  is  not  the  case,  as  when  a  ball  is  struck  upon  its 
side  by  a  bat,  there  is  a  tendency  to  rotate,  and  the  body 
moves  in  a  direction  which  varies  more  or  less  from  that 
of  the  force. 

In  general,  if  any  number  of  forces  act  upon  a  body, 
this  law  will  hold  good  for  each  of  them  as  tliough  the 
others  did  not  exist ;  or  the  effect  upon  the  body,  as  re- 
gards quantity  and  direction  of  motion,  at  the  end  of  any 
time,  is  the  same  as  if  the  forces  had  acted  each  through 
the  same  time  in  succession. 

So  far  as  it  relates  to  the  direction  and  rate  of  motion 
this  is  properly  the  basis  of  the  Parallelogram  of  Veloci- 
ties, explained  in  Art.  33.  For  example,  in  the  cross- 
ing of  the  stream  by  the  uniformly  moving  boat  (3G), 
each  motion  goes  on  independently,  not  modified  by  the 
existence  of  the  other,  although  the  position  of  the  boat 
at  any  moment  depends  on  tlie  action  of  both.  Again, 
every  motion  goes  on  aboard  a  steamboat  indifferently 
whether  the  boat  is  or  is  not  in  motion :  a  ball  thrown 
up  by  a  person  standing  on  deck  falls  again  to  his  hand, 
and  if  dropped  from  the  top  of  the  mast  falls  at  its  foot 
a  second  or  two  after,  although  the  motion  of  the  boat 
may  have  been  rapidly  forward  all  the  time.  So,  too,  a 
rifle-ball  fired  horizontally  from  a  window,  and  another 
dropped  vertically  down  at  the  same  instant,  reach  the 
ground  in  the  same  time  (allowance  being  made  for  the 
resistance  of  the  air).  In  each  case  the  forward  and 
the  downward  motions  go  on  independently  of  each 
other. 

69.  The  THIRD  LAW  is  sometimes  briefly  stated  in  this 
form,  that:  The  actmi  and  reaction  are  equal  and  con- 
trary. 


70  DYNAMICS.  [70. 

The  general  term  stress  is  employed  to  express  the  mutual 
action  between  two  bodies,  for  the  action  of  a  force  always 
implies  two  bodies,  and  this  law  affirms  that  the  action  of  the  one 
is  exactly  equal  to  the  reaction  of  the  other. 

This  law  is  true  whether  the  force  is  exerted  as  pres- 
sure, as  a  pull  or  tension,  or  as  a  blow. 

{a)  Pressure.  For  instance,  when  the  hand  presses 
against  the  wall,  a  contrary  and  equal  reaction  is  exerted 
by  the  wall.  A  weight,  exerting  pressure  on  a  support, 
encounters  an  equal  return  pressure  upward. 

(h)  Tension,  For  example,  a  ten-pound  weight,  hang- 
ing by  a  string,  exerts  a  pull  of  10  lbs.,  and  the  reaction 
of  the  hook  to  which  the  string  is  attached  is  also  10  lbs. 
Also,  if  a  horse  drags  a  heavy  load  by  a  rope,  the  load 
pulls  back  an  amount  equal  to  that  which  the  horse 
exerts. 

{c)  Blow.  When  a  hammer  strikes  a  nail,  the  reaction 
is  equal  and  contrary.  Again,  when  a  cannon  is  fired 
off,  the  recoil  of  the  gun  and  carriage  is  due  to  the 
reaction  equal  to  the  action  in  driving  forward  the  shot. 

The  third  law  of  motion,  taken  in  its  broadest  sense, 
is  true  only  when  the  apparent  loss  of  mechanical  energy 
on  impact  is  allowed  for,  as  explained  later  (108  et  seq.). 

70.  Collision  of  Inelastic  Bodies.  It  follows  from  the 
preceding  article  that:  Whe7i  several  bodies  come  into 
direct  collision,  the  momentum  of  the  whole  system  before 
and  after  impact  is  the  same. 

Suppose  two  inelastic  bodies  whose  masses  are  M  and 
m,  and  whose  velocities  are  V  and  v)  the  momentum  of 
the  first  is  MV,  and  of  the  second  is  7nv.  (a)  If  they 
are  moving  in  the  same  direction,  the  momentum  of  the 
two  is 

MV-}-mv. 


70.]  LAWS   OF  MOTION.  71 

If  now  the  faster-moying  body  overtakes  and  impinges 
upon  the  other,  the  two  after  impact  will  move  along 
together  with  a  velocity  v'  less  than  V  and  greater  than 
v\  the  momentum  of  the  two  together  will  be 

(if  +  m)  v\ 
and  by  this  law 

MY  +  mv  =  (if  +  m)  v\  (1) 

(5)  If  the  two  bodies  were  moving  in  opposite  direc- 
tions, then  the  momentum  of  the  two  moving  separately 

is 

MY  —  mv, 

and  after  impact  of  the  two  together, 

(M  +  m)  v'\ 
and,  as  before, 

MY  -  mv  :=(M-\-m)  v\  (2) 

As  an  example  of  this,  suppose  that  a  rifle-ball  weigh- 
ing one  ounce  and  moving  with  an  unknown  velocity  v 
strikes  and  penetrates  a  body  whose  weight  is  10  lbs., 
and  that  after  impact  the  velocity  of  the  mass  of  wood 
with  the  imbedded  ball  is  v'  =  8  feet  per  second;  then 
A  X  V  =  (10  +  ,1^)  X  8. 

From  which  it  may  be  calculated  that  v  =  1288. 
This  is  the  principle  involved  in  the  lallistic  pendulum; 
the  velocity  after  the  impact  is  determined,  however, 
not  directly  but  by  calculation  from  the  height  to 
which  the  mass  is  raised  {y  =  2g7i). 

If  the  impinging  bodies  are  perfectly  or  imperfectly 
elastic,  the  conditions  are  changed,  and  a  factor  express- 
ing the  degree  of  elasticity  (coefficient  of  elasticity) 
must  be  introduced.  The  discussion  of  these  cases  lies 
outside  of  the  scope  of  the-  present  work. 


72  DYNAMICS.  [71, 

EXAMPLES. 

XIII.  Collision  of  Inelastic  Bodies.    Article  70. 

[The  bodies  are  supposed  to  be  perfectly  inelastic,  and  their  mo- 
tion is  uniform;  the  impact  is  direct,  not  oblique.] 

1.  A  ball  weighing  10  lbs.  and  having  a  velocity  of  16  feet 
per  second  overtakes  a  second  ball  weighing  5  lbs.  and  whose 
velocity  is  8  feet  per  second :  What  is  the  final  velocity  ? 

2.  If  the  first  ball  in  the  preceding  example  meets  the  second, 
what  is  the  final  velocity  ? 

3.  A  body  weighing  40  lbs.  strikes  another  at  rest  weighing  360 
lbs.,  and  the  two  move  on  with  a  velocity  of  2  feet  per  second; 
What  was  the  original  velocity  of  the  first  ball  ? 

4.  Three  bodies,  each  weighing  4  lbs. ,  are  situated  in  a  straight 
line;  a  fourth,  weighing  8  lbs.  and  moving  at  a  rate  of  12  feet  per 
second,  strikes  them  in  succession:  What  velocity  results  after 
each  impact  ? 

5.  Two  bodies  moving  in  the  same  direction  at  the  rates  of  8 
and  10  feet  per  second  come  into  collision,  and  after  impact  have 
a  velocity  of  8.4  feet  per  second:  What  is  the  ratio  of  the  masses 
of  the  two  bodies  ? 

6.  If  the  bodies  in  example  5  move  in  opposite  direction,  and 
the  final  velocity  is  .4  feet  per  second,  what  is  the  ratio  of  their 
masses  ? 

7.  A  body  moving  10  feet  per  second  meets  another  moving 
2  feet  per  second,  and  thus  loses  one  half  of  its  momentum :  What 
is  the  ratio  of  the  masses  of  the  two  bodies  ? 

8.  A  body  weighing  6  lbs.  strikes  another  weighing  5  lbs.  and 
moving  in  the  same  direction  at  a  rate  of  7  feet  per  second :  If  the 
velocity  of  the  second  body  is  doubled  by  the  impact,  what  was 
the  previous  velocity  of  the  first  body? 

9  An  ounce  rifle-bullet  is  fired  (as  in  70)  into  a  suspended  block 
weighing  36  lbs. ;  the  blow  causes  the  wood  to  rise  li  inches :  Re 
quired  the  velocity  of  the  bullet  at  the  moment  of  impact. 

Measurement  of  Force. 
71    Absolute  Method  of  Measuring  Force.     A  force 
may  be  measured:  By  the  velocity  which  it  gives  the  unit 


72.]  MEASUKEMENT  OF  FOECE.  73 

of  mass  in  the  unit  of  time.  The  unit  force  is  tlien  a 
force  which  will  giye  a  pound  of  matter  a  velocity  of  one 
foot  per  second  in  a  second.  This  unit  is  sometimes 
called  a  poundal.  It  may  also  be  stated  in  this  equiva- 
lent form,  already  implied  in  Art.  68:  A  unit  force  is 
one  which  will  generate  (or  destroy)  a  unit  of  momen- 
tum in  one  second. 

"When  the  units  of  the  metric  system  are  employed,  a  unit  force 
is  defined  as  one  which  will  give  one  gram  of  matter  a  velocity  ol 
one  centimeter  per  second  in  one  second ;  this  unit  force  is  called 
a  Dyne.  13,825.38  dynes  make  one  poundal.  This  system  of 
measuring  force  is  called  the  centimeter-gram-second  system,  or 
the  C.G.S.  system. 

The  force  of  gravity  on  a  pound  of  matter,  which  giveb 
a  velocity  of  about  32  feet  per  second  in  a  second  {g),  is 
then  a  force  of  32  poundals,  and  hence  this  number  32 
(or  g)  is  the  measure  of  the  earth's  attraction  on  this 
absolute  system.  As  it  is  32  times  the  force  required  to 
give  one  pound  a  velocity  of  one  foot  per  second,  it  is 
evident  that  the  unit  force,  the  poundal,  is  equivalent 

to  the  action  of  gravity  on  about  half  an  ounce  (-^ry^)  • 

This  method  of  measuring  force  is  called  alsolute 
measure,  in  the  sense  that  it  is  universally  applicable  and 
independent,  as  the  following  method  is  not,  of  the  vari- 
ations in  the  force  of  gravity.  As  implied  above,  the 
UKIT  OF  MASS  in  this  system  is  the  standard  pound. 

72.  Gravitation  Method  of  Measuring  Force.  Forces 
are  also  measured:  By  comparing  them  directly  with 
gravity;  that  is,  by  the  weights  they  could  support.  The 
UNIT  FOKCE  is  tlicu  a  force  equal  to  that  required 
to  support  the  standard  pound  against  the  force  of 
gravity;   or,  briefly,  it  is  equal  to  the   weight   of   one 


74  DYNAMICS.  [72. 

pound.  It  is  then  g  times,  or  about  32  times,  the  unit 
of  force  mentioned  in  the  preceding  article.  This  is 
called  gravitation  measure. 

Since  now  the  force  of  gravity  manifests  itself  every- 
where and  at  all  times  on  the  earth,  and  since  we  are 
so  familiar  with  its  intensity  as  measured  by  the  weights 
of  one,  two,  ten  pounds,  and  the  amount  of  muscular 
exertion  required  to  overcome  it,  this  is  a  most  simple 
and  natural  way  of  measuring  all  forces.  After  this 
method  we  say  that  the  tension  of  the  rope  pulling  a 
canal-boat  is  100  lbs.  when  the  force  exerted  is  equal  to 
that  required  to  support  a  weight  of  100  lbs.  It  is 
common  to  speak  of  a  pull — as  on  an  oar — of  50  lbs.,  of 
the  force  of  the  wind  or  that  of  the  waves  as  being  so 
many  pounds,  etc.  In  cases  like  the  last  a  dynamometer 
is  employed,  and  the  pressure  on  a  spring  noted,  and  this 
readily  compared  with  the  same  effect  produced  by  a 
known  weight  under  the  action  of  gravity. 

Notwithstanding  the  fact  that  this  method  is  so  com- 
monly and  conveniently  employed,  it  is  less  scientific 
than  the  absolute  method,  and  is  open  to  one  serious 
objection,  that  it  does  not  necessarily  take  into  account 
the  variations  in  the  force  of  gravity.  As  explained  in 
Art.  64,  the  force  of  gravity  varies  about  -g-f^  between 
the  equator  and  the  poles,  and,  when  the  same  mass  is 
used  as  the  unit  of  weight,  a  pull  of  a  pound  means  a 
stronger  pull  in  high  latitudes  than  toward  the  equator. 
Hence  the  gravitation  method  is  accurate  only  when  the 
difference  in  the  value  of  g  at  the  spot  in  question  an  d 
at  the  sea-level  is  known  and  taken  into  account. 

When  forces  are  considered  as  producing  accelerated 
motion,  the  absolute  measure  is  generally  employed;  but 
when  they  act,  as  usually  in  Statics,  either  as  tensions  or 


73.]  DYNAMICAL  PROBLEMS.  75 

pressures,  the  gravitation  measure  is  the  one  usually 
accepted. 

The  UN^iT  OF  MASS  employed  (in  the  gravitation  meas- 
ure of  force)  is  the  quantity  of  matter  in  a  body  which 
weighs  g  pounds,  where  g  is  the  acceleration  of  gravity 
for  the  place  in  question.  This  is  then  a  varying  unit, 
having  a  definite  value  for  each  place  under  considera- 
tion. The  reason  for  selecting  this  unit  is  that  the 
numerical  expression  of  the  mass  of  a  given  body  will  be 
the  same  everywhere;  that  is,  a  body  of  mass  10  will  still 
be  this  wherever  it  is.    For  since,  as  has  been  shown  (68), 

W  W 

we  may  take  M  =  — ,  therefore  this  ratio  of  — ,  and 

^  9  9 

hence  the  numerical  value  of  if,  will  be  always  the  same. 
For  example,  on  the  sun,  where  the  force  of  attraction  is 
about  28  times  that  on  the  earth,  we  should  have 

i^r       28.  W 

or  the  same  value  as  before. 

ProUems  in  Dynamics. 

73.  In  Art.  68  it  was  shown  that  the  intensity  of  any 
acting  force  {F)  is  equal  to  the  product  of  the  mass 
moved  {M)  into  the  velocity  generated  in  one  second; 

that  is, 

F  =  Mf.  (1) 

/     W\ 
If  in  this  equation  we  substitute  the  value  of  if  I  = — , 

W 
we  obtain  F=  —  ,f,  (2) 

pr  '  f=^-ff-  (3) 


76 


DYNAMICS. 


[74. 


This  relation  makes  it  possible  to  obtain  the  accelera- 
tion (/)  produced  by  any  known  force  acting  upon  a 

body  whose  weight  is  also 
known.  When  /  is  known, 
tlien  the  equations  of  articles 
27,  41,  42  give  the  means  of 
calculating  all  the  particulars 
in  regard  to  the  motion  of  the 
body;  that  is,  the  space  passed 
through  in  a  given  time,  the 
velocity  acquired,  etc. 

74.  Attwood's  Machine. 

Equation  (3)  in  the  preced- 
ing article,  in  connection 
with  Attwood's  macliine, 
makes  it  possible  to  verify 
the  laws  of  motion  given  in 
articles  67,  68.  The  essen- 
tial parts  of  the  machine  are 
shown  in  Fig.  29.  There  is 
a  pulley  over  which  a  thread 
passes,  holding  any  weights 
P  and  Q.  The  axis  of  the 
pulley  rests  on  two  smaller 
wheels,  called  friction-iolieels, 
which  serve  to  diminish  the 
friction  so  that  its  effect  can  be 
disregarded.  The  whole  is  sup- 
ported by  a  firm, massive  stand. 
In  the  path  of  the  weights 
maybe  clamped  a  stage  (as  d) 
at  any  point,  as  determined  by 
Fio,  29.  V     the  vertical  scale/).  A  ring  {c) 


74.]  DYNAMICAL  PROBLEMS.  77 

in  the  same  position  seryes,  if  desired,  to  remove  a  por- 
tion of  the  weight  without  disturbing  the  motion  of  the 
rest.  For  this  last  purpose  the  extra  weight  is  in  the 
form  of  a  straight  bar  (^),  which  is  caught  by  the  ring 
as  the  weight  moves  through  it.  A  pendulum  to  beat 
seconds  is  added  to  the  whole. 

The  following  are  some  of  the  experiments  which  may 
be  tried: 

(1)  Suppose  the  weight  P  =  8J  oz.  and  §  =  7f  oz., 
supported  on  either  thread-;  then  the  weight  moved  ( W) 
is  =  P  +  g  =  16  oz.  (=  1  lb.),  and  the  moving  force  (P) 
is  the  difference,  or  P  —  Q  =  i  oz.  [Strictly  the  weight 
moved  also  includes  the  weight  of  the  pulley,  which  in 
an  accurate  experiment  would  have  to  be  taken  into 
account.]     Therefore,  by  equation  (3),  if  ^  =  32, 

f=  —-.g  =  — ^.g  =  1  foot-per-second  per  second. 

Now,  by  Art.  27,  the  space  passed  through  in  the 
first  two  seconds  {t  =  2)  by  a  body  moving  with  uni- 
formly accelerated  motion  is  2/.  Clamp  the  stage  at  a 
distance  of  2  feet  below  the  initial  point  of  P,  and  the 
weight  descending  will  strike  the  stage  exactly  two 
seconds  after  starting.  This  is  also  a  verification  by 
experiment  of  the  remark,  made  in  Art.  71,  that  the 
unit  force,  the  poundal — that  is,  the  force  which  would 
give  the  unit  of  mass  1  lb.  (here  F  +  Q)  m  a  second  a 
velocity  of  one  foot  per  second — was  equal  to  the  weight 
ol  i  oz.  (=  P-Q). 

If  the  stage  be  clamped  at  4:^  feet,  then  the  weight 
will  strike  it  at  the  end  of  three  seconds,  as  the  formula 
s  =  ^ff  requires.  Showing,  too,  that  in  three  seconds 
the  space  passed  through  is  f  times  that  in  two  seconds; 


78  DYNAMICS.  [74. 

or,  in  other  words  (27),  the  space  is  ^proportional  to 
the  square  of  the  time  (2''  :  3"  =  4  :  9). 

(2)  Again,  let  P  =  8.5  oz.  and  §  =  7.5  oz.;  then 
P  +  5  =  16  oz.,  or  the  weight  moved  is  the  same  as  in 
(1),  but  the  moving  force  P  —  §  =  1  oz.,  or  twice  that 
in  (1);  then,  as  before, 

P  —  0 

f  =  p      ^ '  ff  =^  f  eet-per-second  per  second. 

Hence,  in  accordance  with  the  law  stated  in  Art.  68, 
the  mass  remaining  constant,  the  velocity  generated  in  a 
given  time  is  proportional  to  the  force  acting;  that  is, 

Z.  -  /  _  1 

P'  -  /  -  2' 

If,  as  before,  the  stage  is  clamped  4  feet  below  the 
starting-point,  the  weight  will  strike  it  at  the  end  of  two 
seconds,  as  the  formula  requires  (s  =  iff). 

(3)  LetP  =  4ioz.  and  §  =  3f  oz.;thenP+ §  =  8  oz.; 
that  is,  the  weight  (or  mass)  moved  is  one  half  that  in 
(1),  while  P  —  Q  =  i  oz.;  that  is,  the  moving  force  is 
the  same.     Equation  (3)  gives 

P  —  Q 
f  =  p  ,Q  ff  =  ^  f  eet-per-second  per  second. 

That  is,  in  accordance  with  the  law  stated  in  Art.  68, 
the  acting  force  being  constant,  the  velocity  generated  i^i 
a  given  time  is  inversely  as  the  masses  acted  upon» 

/  _  fi  _  _16  _  ^ 
/  ~    m'  ~    8   ""  1' 

This  value  of/  may  be  verified  as  before. 

(4)  Let  P  =  8.5  oz.  and  §  =  7.5  oz.,  and  let  the 
excess  of  P  over  Q  (1  oz.)  be  in  the  form  of  a  rod  pro- 


75.]  DYNAMICAL  PROBLEMS.  79 

jecting  so  as  to  be  removed  by  the  ring.  Then,  for  this 
case,/  =  2;  hence  if  the  ring  be  placed  one  foot  below 
the  starting-point,  the  weight  will  reach  it  at  the  end  of 
one  second.  Here  the  extra  weight  is  left  behind,  and 
now,  the  two  weights  being  equal,  the  bodies  must, 
according  to  the  first  law  of  motion,  move  on  with  uni- 
form velocity.  At  the  end  of  another  second  it  will 
strike  the  stage  if  placed  2  feet  below — that  is,  3  feet  from 
the  starting-point — and  at  the  end  of  two  seconds  at  5, 
and  so  on. 

75.  The  following  is  a  similar  application  of  the 
above  principle.    Let  W  (Fig.  jy 

30)  be  a  weight  resting  on  a      ^^  - 

perfectly    smooth    horizontal     ^ 

plane;  P  is  another  weight 

attached  to    TF  by  a    string 

passing  over    the    pulley   a; 

then,   neglecting  the  weight  fio.  3o. 

of  the  pulley,  when  P  falls  it  moves  W  also;  hence  the 

total  weight  moved  is  P  +  W,  and  the  moving  force  is 

P,     Hence  equation  (3)  becomes 


/  = 


P  -h  W 


The  tension  (T)  of  the  string  is  given  from  the  rela- 
tion 

I  __/ 

substituting  in  this  the  above  value  of/,  we  obtain 

PW 


80  DYKAMICS.  [7a 

76.  The  relation  (3)  in  Art.  73  is  applicable  to  a  re- 
tarding force  sucli  as  friction.  For  example,  let  Who  a, 
weight  moving  on  a  rough  horizontal  plane;  the  force  of 
a  friction  (F)  is  a  force  acting  parallel  to  the  surface 
in  a  direction  opposite  to  the  motion.  The  retardation 
due  to  friction  is  given  by  the  equation 

If  F  is  known,  and  also  W,  then  /  can  be  calculated; 
this  is  the  retardation  due  to  friction;  in  other  words, 
the  body  loses  each  second  in  velocity  /  feet  per  sec- 
ond. The  distance  which  the  body  will  go  through 
before  coming  to  rest,  and  its  distance  at  any  moment 
from  the  starting-point,  will  be  given  by  the  formulas 

r=u-ft, 

s  =  ut-  ift\ 
Other  illustrations  are  given  in  the  examples  below. 

XIV.  General  Dynamical  Problems.    Articles  68,  73-76. 

1.  If  in  Attwood's  machine  P  =  4^  oz.  and  Q  =  3^  oz. :  {a)  What 
is  the  acceleration?  (6)  What  space  will  be  passed  through  in 
2  seconds? 

2.  {a)  At  what  height  above  the  earth's  surface  would  a  body- 
fall  4  feet  in  the  first  second  from  rest?  (5)  If  its  weight  was  40 
lbs.,  what  pull  would  it  exert  on  a  spring-balance  at  this  point? 

3.  A  12-lb.  weight  hanging  over  the  edge  of  a  smooth  table 
drags  a  60-lb.  weight  with  it:  What  is  the  acceleration  and  the 
tension  of  the  string? 

4.  If  the  table  in  example  3  is  rough,  and  the  resistance  of 
friction  consequently  equivalent  to  one  tenth  of  the  weight  of  the 
sliding  body,  what  is  the  acceleration? 

5.  {a)  A  bucket  weighing  100  lbs.  is  raised  up  from  a  well  at 


76.]  DYNAMICAL  PEOBLEMS.  81 

a  uniform  rate  of  12  feet  per  second :  What  is  the  tension  of  the 
rope?  (b)  If  the  acceleration  is  4  feet-per-second  per  second, 
what  is  the  tension? 

6.  For  what  time  must  a  force  of  4  oz.  (gravitation  measure) 
act  on  a  body  weigliing  8  lbs.  to  give  it  a  velocity  of  20  feet  per 
second? 

7.  Two  weights  of  16  and  14  oz.  hang  over  a  pulley:  What 
space  will  they  move  through  from  rest  in  3  seconds? 

8.  Two  weights,  each  8  oz.,  hang  over  a  pulley:  What  addi- 
tional weight  must  be  added  to  one  of  them  to  give  an  accelera- 
tion of  2  feet-per-second  per  second? 

9.  A  weight  of  8  lbs.  rests  on  a  smooth  horizontal  table  12  feet 
wide:  What  weight  hanging  vertically  will  draw  it  across  in 
3  seconds? 

10.  A  weight  of  24  lbs.  rests  on  a  platform:  (a)  What  is  its 
pressure  on  the  platform  if  the  latter  is  ascending  with  an  accele- 
ration of  ig^    (b)  If  descending  with  the  same  acceleration? 

11.  A  body  weighing  160  lbs.  is  moved  by  a  constant  force, 
which  generates  a  velocity  of  8  feet  per  second :  What  weight 
could  the  force  support? 

12.  A  force  of  8  lbs.  (gravitation  measure)  acts  constantly  on  a 
body  weighing  24  lbs.  and  resting  on  a  smooth  horizontal  sur- 
face :  (a)  What  is  the  acceleration,  and  (b)  how  far  will  the  body 
move  in  4  seconds? 

13.  The  velocity  of  a  body  weighing  24  oz.  is  increased  from 
20  to  40  feet  per  second  while  the  body  passes  over  30  feet:  What 
is  the  moving  force? 

14.  Of  two  weights  hanging  over  a  pulley,  one  is  1  lb.  and  it 
ascends  with  an  acceleration  of  10  feet-per-second  per  second: 
What  is  the  other  weight? 

15.  How  long  must  a  constant  force  of  10  lbs.  act  on  a  mass  of 
100  lbs.  to  give  it  a  velocity  of  30  miles  an  hour? 

16.  What  constant  force  will  cause  a  body  weighing  400  lbs.  to 
pass  over  1200  feet  in  10  seconds  from  rest  on  a  smooth  horizontal 
surface? 

17.  A  constant  force  of  15  lbs.  gives  a  body  an  acceleration  of 
5  feet  per  second  in  one  second:  What  is  the  weight  of  the  body? 


18.  A  body  weighing  24  lbs.  is  projected  on  a  rough  horizontal 


82  DYKAMICS.  [76. 

surface,  where  the  resistance  of  friction  is  6  lbs.,  with  an  initial 
velocity  of  64  feet  per  second:  {a)  How  far  and  (p)  how  long  will 
it  slide  before  coming  to  rest? 

19  A  body  weighing  40  lbs,  is  projected  as  in  the  preceding 
example.  What  is  the  resistance  of  friction  if  the  body  slides 
16  seconds  before  stopping? 

20  A  body  weighing  60  lbs.  is  projected  up  a  rough  plane  in- 
clined at  an  angle  of  30°,  where  the  resistance  of  friction  is  6  lbs,, 
and  with  an  initial  velocity  of  160  feet  per  second:  {a)  How  far 
on  the  plane  will  it  ascend?    (6)  How  long  will  it  take? 

21.  The  length  of  an  inclined  plane  is  1000  feet,  and  its  base 
800  feet;  the  resistance  of  friction  is  one  eighth  of  the  weight: 
{a)  What  initial  velocity  must  it  have  just  to  reach  the  top,  and 
(b)  how  long  will  the  ascent  take  ? 


CHAPTER  HI.— CENTRAL  FORCES. 

77.  Uniform  Circular  Motion.  Suppose  a  body  to  be 
moving  in  a  circular  path  with,  a  constant  velocity ; 
according  to  the  first  law  of  motion  (66),  a  body  in 
motion,  if  not  acted  upon  by  any  force,  tends  to  move 
on  uniformly  in  a  straight  line.  In  order,  therefore, 
that  this  body  should,  as  supposed,  move  in  a  circle,  it 
must  be  acted  upon  by  a  constant  force  exerted  in  the 
direction  of  the  centre  of  the  circle. 

This  force  toward  the  centre,  or  central  force,  is  called 
the  centripetal  force.  The  equal  and  opposite  (69) 
reaction  exerted  away  from  the  centre  is  called  the 
centrifugal  force.  The  central  forces  determine  the 
direction  of  motion  of  the  body,  but  do  not  affect  its 
rate  of  motion  or  velocity,  since  they  act  continually  at 
right  angles  to  its  path.  If  a  body  attached  to  a  string 
be  whirled  about  a  centre,  the  intensity  of  these  cen- 
tral forces  is  measured  by  the  tension  of  the  string. 
If  the  string  be  cut,  the  body  will  fly  off  in  a  tangent  to 
the  curve,  but  with  unchanged  velocity. 

78.  To  find  the  intensity  of  the  central  force  in  the 
case  of  uniform  circular  motion  :  Suppose  the  body  to 
be  moving  at  a  constant  velocity  v  about  the  c\yc\qADH 
(Fig.  31),  whose  radius  {AC)  is  r.  In  a  very  short  space 
of  time  {t)  it  will  have  gone  from  A  to  D,  so  that 

AD  =  vt.  (1) 


84 


DYNAMICS. 


[78, 


But  in  this  time  it  has  been  drawn  away  from  the  tan- 
gent toward  the  centre  a  distance  equal  to  BD  (oyAB), 
Let /be  the  acceleration  of  the  constant  central  force; 
then  (60) 

AB  =  iff.  (2) 

But  since  AFH  is  a  semicircle,  the  angle  ADH  is  a  right 
angle,  and,  by  geometry, 


ad'  =  AE  X  AH=  AEx  2r, 


(3) 


If  AD  IS  taken  very  small,  the  chord  AD  may  be 
regarded  as  identical  with  the  arc  AD.      Therefore, 


introducing  into  (3)  the  values  given  in  (1)  and  (2),  we 
have 

v'f  =  iff  X  2r, 


or 


(4) 


This  value  of  /  (4)  gives  the  acceleration  due  to  the 
constant  force  in  terms  of  the  velocity  and  the  radius  of 
the  circle.     From  it  we  see  that  the  value  of  f  varies 


79.]  CENTRAL  FORCES.  85 

directly  with  the  square  of  the  Telocity,  and  inversely  as 
the  radius. 

In  order  to  obtain  the  intensity  of  the  force  (F) — that 
is,  in  the  case  supposed  above,  the  tension  of  the  string — 
the  value  of /must  be  multiplied,  as  explained  in  68,  by 
the  mass  (M)  of  the  body  in  motion.     "We  have  then 

If  the  value  of  F  in  pounds  be  required,  when  the 

W 
weight  (W)  is  given:  since  M  =  — ,  we  have,  further, 

g     r 

79.  The  pull  away  from  the  centre,  called  the  cen- 
trifugal force,  is  felt  whenever  a  body  is  made  to  rotate 
rapidly  about  a  fixed  centre.  It  is  exemplified  by  the 
case  of  a  loaded  sling:  if  the  cord  is  elastic,  the  extent 
to  which  it  is  stretched  is  a  measure  of  this  force.  Simi- 
larly, a  bucket  containing  water  may  be  swung  around 
by  a  rope  so  rapidly  that  this  force  becomes  greater  than 
that  of  gravity,  and  the  contents  are  consequently  not 
lost  even  when  it  is  inverted. 

In  the  case  of  a  large  wheel  rotating  rapidly,  if  the 
centre  of  gravity  and  the  axis  of  rotation  coincide,  the 
effects  upon  the  parts  on  opposite  sides  of  the  axis  neu- 
tralize each  other  and  produce  no  result,  except  when 
the  force  becomes  greater  than  the  cohesion  between 
the  particles,  when  fracture  takes  place — as  when  a 
grindstone  breaks.  If,  however,  the  centre  of  gravity 
does  not  coincide  with  the  axis,  a  continuous  pull  on  the 
bearing  is  produced  by  the  motion  which  may  lead  to 


86  DYNAMICS.  [80. 

Tery  in  jiirious  results.  For  the  same  reason  a  crank-arm, 
which  in  use  is  turned  rapidly,  is  generally  weighted  on 
the  side  of  the  centre  opposite  to  the  handle,  to  neutral- 
ize the  injurious  pull  on  the  axis  that  would  otherwise 
exist. 

If  a  globe  containing  a  little  mercury  be  set  in  rapid 
rotation,  the  effect  is  to  cause  the  mercury  to  recede 
from  the  axis  and  hence  to  rise  and  form  a  ring  about 
the  central  part  farthest  from  the  axis.  The  gOYcrnor 
of  Watt,  applied  to  the  steam-engine,  consists  essen- 
tially of  two  heayy  balls  carried  on  rods  jointed  at  the 
top.  They  are  connected  with  some  turning  part  of  the 
engine,  and,  on  the  above  principles,  an  increase  in  the 
rate  of  revolution  causes  them  to  separate  and  rise,  and 
conversely  if  the  rate  is  diminished.  The  arrangement 
IS  such  that  in  the  former  case  they  partially  close,  and 
in  the  other  case  open,  a  valve  by  which  the  supply  of 
steam  is  received.  They  thus  serve  to  regulate  the 
motion  and  keep  it  uniform;  whence  the  name  the 
instrument  has  received. 

80.  Centrifugal  Force  due  to  the  Earth's  Rotation. 

Since  the  earth  is  rotating  on  its  axis,  every  body  on  its 
surface,  tending  to  move  on  in  a  straight  line,  must  be 
retained  there  by  a  force  pulling  toward  the  axis.  In 
other  words,  a  certain  portion  of  the  earth's  attraction 
for  every  body  on  its  surface  is  exerted  simply  to  con- 
strain the  body  to  move  in  a  circle,  and  is  consequently 
not  felt  as  weight.  This  is  equivalent  to  saying  that  the 
centrifugal  force  acts  on  every  body  directly  away  from 
the  centre  of  the  circle  in  which  it  is  moving.  The  di- 
rection of  this  force  is  indicated  for  the  points  E  and  B 
(Fig.  32)  by  the  lines  EG  and  BK, 


81]. 


CENTEAL   FORCES. 


87 


Since,  by  the  preceding  article, 

the  value   of  /  can  be  calculated  for  the  equator,  for 
the  value  of  v  is  given  by  the  fact  that  any  point  on  it 


-of^ — ^ — 

r 


Fio.  32. 

describes  a  distance  equal  to  the  equatorial  circumference 
in  24  hours.     In  this  way  we  obtain 

/  =  .1112  feet-per-second  per  second. 

This  value  of /is  about  -^  of  the  value  which  g  would 
have  at  the  equator  if  this  influence  did  not  exist;  since 
17''  zz:  289,  it  follows  that,  if  the  velocity  of  rotation  of 
the  earth  were  increased  17  times,  all  bodies  upon  it 
at  the  equator  would  entirely  lose  their  weight. 

81.  For  any  point  as  B  (Fig.  32)  where  the  latitude 
is  I,  the  value  of  this  force  /',  and  the  velocity  v'  (=  '» 
cos  l\  we  have 

^        v'^       v^  cos'  I       v^        ^ 

/  =   — 7"  = J  =    -  cos  ly 

-'  r  r  cos  I         r 

.-. /=/cos?. 


88  DYl^AMICS.  [81. 

The  components  of  /'  {BK,  Fig.  31)  are  BH  normal 

to  the  surface  and  BL  as  a  tangent.     Of  these,  since 

HBK  =  Z, 

BR  =  f  cosl=f  cos' I; 

sin  21 
BL  =  /'  sin  Z  =  /  sin  I  cos  I  =  f       ^ 

The  normal  component  alone  influences  the  weight  of 
the  body  as  it  acts  directly  contrary  to  gravity,  while  the 
tangential  component  tends  to  produce  motion  toward 
the  equator.  It  was  the  influence  of  the  tangential  com- 
ponent when  the  earth  was  in  a  plastic  condition  which 
is  belieyed  to  have  caused  the  flattening  at  the  poles. 
This  effect  may  be  illustrated  by  the  rapid  rotation  of  a 
plastic  mass  of  clay  on  its  axis. 

The  yalue  of/ is  greatest  at  the  equator  and  dimin- 
ishes, with  the  cosine  of  the  latitude,  as  we  go  toward 
the  poles;  at  the  poles  it  is  zero  (cos  I  =  cos  90°  =  0). 
The  normal  component  is  gi'eatest  at  the  equator  and 
diminishes  with  the  square  of  cosine  of  the  latitude. 
The  tangential  component  is  zero  at  the  equator,  in- 
creases to  latitude  45°  (^Z=-^  j  and  diminishes  from 
there  to  the  poles,  where  it  is  again  zero. 

EXAMPLES. 
XV.  Centripetal  and  Centrifugal  Forcss.    Articles  77-81. 

1.  A  ball  weighing  20  lbs.  is  whirled  by  means  of  a  string 
around  a  centre  at  a  radius  of  7  feet,  with  a  linear  velocity  of  28 
feet  per  second-  What  is  the  value  of/,  and  what  is  the  tension  of 
the  string  {F)  ? 

2.  (a)  If  the  velocity  is  doubled  in  the  preceding  ex-ample,  what 
do  the  values  of /and  i^  become  ?  ib)  What  are  they  if  the  radius 
Is  doubled  ? 


81.]  CENTRAL   FORCES.  89 

3.  A  ball  weighing  4  lbs.  attached  to  a  centre  at  a  distance  of  8 
feet  makes  300  revolutions  in  a  minute :  What  is  the  pull  on  the 
centre  ? 

4.  What  linear  velocity  of  rotation  must  a  body  have  if  the 
tension  of  the  string  by  which  it  is  attached  to  the  centre,  at  a 
distance  of  8  feet,  is  equal  to  its  weight  ? 

5.  What  is  the  angular  velocity  of  a  body  moving  in  a  circle 
with  a  radius  of  4  feet,  when  the  centrifugal  force  is  one  half  the 
weight  ? 

6.  A  locomotive  weighing  12  tons  moves  at  a  rate  of  30  miles 
an  hour  about  a  curve  whose  radius  is  1000  feet:  What  is  the 
horizontal  pressure  on  the  rails  ? 

7.  If  a  stone  weighing  5  lbs.  is  attached  to  a  string  3  feet  long 
and  makes  two  revolutions  in  a  second,  what  is  the  pull  on  the 
centre  ? 

8.  If  a  string  can  just  support  a  weight  of  400  lbs.,  what  is  the 
greatest  length  that  can  be  employed  to  swing  around  a  30-lb. 
weight  once  in  a  second  ? 

9.  What  is  the  shortest  length  of  the  string  only  strong  enough 
to  support  100  lbs.  that  can  be  used  to  whirl  around  a  50-lb. 
weight  at  a  rate  of  8  feet  per  second  ? 


CHAPTER  IV.— FRICTIOK 

82.  Definition  of  Friction.  Friction  is  the  resistance 
wliicli  is  offered  to  the  motion  of  one  lody  upon  another 
due  to  the  roughness  of  the  surfaces  i7i  contact.  Fric- 
tion always  acts  parallel  to  the  surfaces,  and  in  a  direc- 
tion contrary  to  that  in  which  the  body  is  moving  or  is 
about  to  move. 

83.  Reaction  of  Smooth  Surfaces.  The  only  effect 
produced  by  the  mutual  pressure  of  two  perfectly  smooth 
surfaces  would  be  the  reaction  perpendicular  to  them  at 
the  point  of  contact.  It  would  hence  exert  no  influence 
on  the  motion  of  one  upon  the  other;  in  such  a  case 
there  would  be  no  friction.  (There  would  still  be,  how- 
ever, even  in  this  case,  resistance  to  motion  due  to  the 
mutual  adhesion  of  the  surfaces  in  contact;  but  this  is 
an  independent  matter  not  here  considered. ) 

An  ideally  smooth  surface  cannot  be  obtained,  even 
by  continued  polishing;  hence  the  resistance  of  friction 
can  never  be  entirely  eliminated.  It  is  found  in  general 
that  the  smoother  the  surface  is  made  the  less  is  the 
friction. 

It  is  obvious,  also,  that  the  actual  reaction  between 
two  rough  surfaces  which  are  in  motion,  or  about  to 
move,  is  in  the  direction  of  the  resultant  of  the  two 
components,  one  the  force  of  friction  parallel  to  the  sur- 
face, and  the  other  the  normal  pressure  as  defined  in 
Art.  89. 


85.]  FRICTION  DEFINED.  91 

84.  Since  friction  is  diminished  by  rendering  the  sur- 
faces in  contact  more  smooth,  it  is  customary  to  make 
use  of  lubricators,  which  fill  up  the  uneyennesses  of  the 
surfaces.  For  example,  oils,  lard,  graphite,  soapstone, 
and  other  substances  are  employed.  The  best  lubri- 
cator, in  a  given  case,  depends  upon  the  materials  in 
contact  and  the  amount  of  pressure  sustained,  and  is 
determined  by  experiment.  The  friction  of  an  axle 
may  be  much  diminished  by  the  use  of  friction-wheels; 
the  axle  rests  upon  two  wheels,  which  turn  with  it,  as 
indicated  in  Fig.  29  (p.  76)  and  Fig.  44  (p.  117). 

Conversely,  when  it  is  desirable  to  increase  friction 
the  surfaces  may  be  made  more  rough.  For  example, 
when  the  driving-wheels  of  a  locomotive  tend  to  slide 
on  the  rails  because  the  latter  are  wet  and  slippery,  it  is 
customary  to  feed  down  sand,  by  a  tube  from  the  sand- 
box above,  on  to  the  rail  in  front  of  each  wheel,  which 
has  the  desired  effect. 

85.  Friction  is,  so  far  as  this,  a  disadvantage  from  a 
mechanical  point  of  view,  since  force  is  required  to  over- 
come it,  and  this  represents  so  much  working  power  or 
energy  expended  without  any  useful  result.  For  ex- 
ample, in  the  machinery  of  a  cotton-mill,  at  every  bear- 
ing there  is  friction,  and  the  engine  which  supplies  the 
energy  for  the  establishment  must  furnish  beyond  what 
is  required  for  the  useful  work  performed  enough  more 
to  make  good  this  waste.  Again,  the  locomotive  on  a 
railroad,  after  the  train  is  in  motion  and  supposing  the 
track  horizontal,  exerts  its  energy  solely  against  the  out- 
side resistance,  which  is  chiefly  caused  by  the  friction  of 
the  axles  in  their  bearings  and  the  wheels  on  the  rails. 

On  the  other  hand,  however,  friction  is  often  mechani- 


92  DYNAMICS.  [86. 

cally  an  advantage.  It  alone  gives  to  tlie  driving-wheels 
of  tlie  locomotive  their  hold  on  the  track;  without  it 
the  belts  in  a  machine-shop  could  not  be  used  to  transmit 
the  motion  of  one  shaft  to  another;  many  mechanical 
arrangements  depend  for  their  efficiency  upon  it.  Even 
so  simple  a  matter  as  walking  would  be  impossible  but 
for  the  hold  on  the  ground  given  to  the  feet  by  friction. 

86.  Kinds  of  Friction.  An  important  distinction  is 
to  be  made  between  sliding  and  rolling  friction.  The 
former  exists  where  there  is  simply  sliding  motion,  as  in 
the  case  of  a  sled,  or  of  an  axle  in  its  bearing.  The 
latter  exists  where  motion  is  accomplished  through  the 
intervention  of  a  wheel  or  roller. 

Sliding  friction  is  that  which  is  especially  investigated 
here.  The  resistance  due  to  it  is  much  greater  than 
that  caused  by  rolling  friction.  This  is  seen  by  the 
effect  produced  when  a  carriage-wheel,  by  a  shoe  or 
some  other  contrivance,  is  made  to  slide  instead  of  roll 
on  the  ground.  The  advantage  of  a  wheel  consists,  as 
respects  friction,  in  this,  that  instead  of  sliding  friction 
on  the  ground,  the  rolling  friction  there  and  the  sliding 
friction  on  the  axle  are  substituted;  the  latter  element 
is  of  comparatively  small  moment.  Similarly,  when 
heavy  weights  are  to  be  moved  for  small  distances, 
rollers  of  iron  or  wood  are  often  placed  under  them. 

87.  Fluid  Friction.  Friction,  or  resistance  to  motion, 
is  also  felt  in  the  case  of  a  liquid  or  gas,  and  is  then 
C'diledi  fluid  friction.  The  resistance  exists  both  between 
the  molecules  of  the  fluid  itself  and  between  them  and 
the  walls  of  the  containing  vessel.  This  is  true,  for 
example,  where  water  passes  through  pipes.  This  por- 
tion of  the  subject  does  not  fall  within  the  scope  of  this 


>.] 


LAWS   OF  FRICTION. 


93 


work,  but  tlie  fact  must  be  noted  with  reference  to  a 
subsequent  article  (104). 

88.  Laws  of  Friction.  Experiments  upon  friction 
have  established  the  following  laws,  which  hold  good  for 
any  two  given  surfaces  in  contact.  It  is  supposed  that 
no  abrasion  of  these  surfaces  takes  place. 

(1)  The  force  of  friction  is  proportional  to  the  normal 
pressure  of  the  surfaces  in  contact. 

(2)  Friction  is  independent  of  the  extent  of  surface 
in  contact  when  the  normal  pressure  remains  the  same. 

(3)  Friction  is  independent  of  the  velocity  of  the 
motion  when  sliding  friction  is  considered. 

89.  (a)  The  normal,  i.e.  perpendicular,  pressure  (E), 
mentioned  in  the  first  law,  is  equal  to  the  weight  of  the 


Fig.  33. 


Fio.34. 


body  if  it  rests  upon  a  horizontal  plane.    Hence  (Fig.  33) 

ll  =  W. 

(b)  If  the  body  rests  upon  an  inclined  plane,  the 
normal  pressure  is  equal  to  that  portion  or  component  of 
the  weight  which  is  perpendicular  to  the  surface.  That 
is  (Fig.  34), 

E  =  W  cos  a. 

For  if  (Fig.  34)  ac  represents  the  weight  of  the  body, 
then  ad  and  ab  are  the  two  partial  forces,  or  compo^ 


94 


DYNAMICS. 


[90. 


nentSy  respectively  parallel  and  perpendicular  to  the 
plane,  into  which  it  may  be  resolved  (analogous  to  the 
resolution  of  velocities,  Art.  38,  p.  28,  or  see  Art.  138, 
p.  144).  Of  these  components,  since  lac  —  HLK—  a, 
.  •.  ad-=  W  sin  a,  and  this  represents  the  force  tending 
to  make  the  body  slide  down  the  plane.  Also,  ab,  i.e. 
^Fcos  a,  is  the  pressure,  normal  to  the  plane,  or  the 
reaction  of  the  plane  as  stated  above. 

(c)  If  a  force  F  acts  on  the  body,  at  an  angle  j3,  tend- 
ing to  make  it  slide  along,  then  the  normal  pressure  is 
increased  (or  diminished)  by  that  component  of  the 
force  acting  at  right  angles  to  the  plane.     If  (Fig.  35) 


J? 

^1 


Fia.  35. 


^\ 


^3 


Fia.  36. 


P  =  CAj  exerted  as  a  push,  then  DA  =  P  sin  y5,  and 

R=z  r  +  P  sin  /?. 

If  (Fig.  36)  P  =  AC,  exerted  as  a  pull,  then  AD  = 
P  sin  /?,  and 

R=  If  -  P  sin  /?. 

In  either  case,  in  order  that  the  body  shall  be  just  at 
the  point  of  moving,  BA  (Fig.  35)  ox  AB  (Fig.  36),  i.e. 
P  cos  /3,  must  be  just  equal  to  the  opposing  force  of 
friction. 


90.  Explanation  of  the  Laws  of  Friction.  (1)  The 
FIRST  LAW  simply  states  that  as  this  normal  pressure 
(defined  in  a,  l,  c  of  the  preceding  article)  is  increased  oi 


L 


91.]  COEFFICIENT  OF  FRICTION.  95 

diminished,  the  resistance  of  friction  increases  or  dimin- 
ishes in  the  same  ratio.  This  law  can  be  demonstrated 
experimentally  by  an  arrangement  like  that  in  Fig.  37. 
The  weight  W  rests  on  a  horizon-  ^  ?r 
tal  plane  (so  that  E  =  TT  as  in  a),  -^= 
A  thread  fastened  to  W  passes 
over  a  pulley  a,  and  is  attached  at 
the  other  end  to  a  second  weight 
P.     In  order  that  TT  shall  be  on  Fio.37. 

the  point  of  moving,  P  must  be  just  equal  to  the  force 
of  friction  (F).  If  W  be  doubled  it  will  be  found  that, 
to  satisfy  the  above  condition,  P  must  also  be  doubled, 
and  so  on.  In  other  words,  the  ratio  ot  P  (=  F)  to 
W  {=  E)  will  remain  constant;  that  is,  the  force  of  fric- 
tion is  proportional  to  the  normal  pressure. 

(2)  The  SECOND  law  may  be  illustrated  by  the  case 
of  a  brick:  supposing  all  the  surfaces  are  alike  in  rough- 
ness, the  friction  is  found  to  be  the  same  upon  which- 
ever of  the  three  surfaces  it  rests. 

(3)  The  THIRD  LAW  is  generally  but  not  rigidly  true. 
It  is  found  (1)  that  in  the  case  of  sliding  friction  the 
resistance  to  motion  is  a  little  greater  when  the  body  is 
just  about  to  move,  and  (2)  that  in  the  case  of  very  high 
velocities  the  friction  becomes  sensibly  diminished. 

91.  Coefficient  of  Friction.  The  constant  ratio  between 
the  force  of  friction  for  two  given  surfaces  and  the  normal 
pressure  is  called  their  coefficient  of  friction,  II  the 
coefficient  of  friction  be  represented  by  /^,  then 

F 
and  F  —  iiE, 


96 


DYJS^AMICS. 


[98 


This  relation  becomes  on  a  horizontal  plane 
F 

on  an  inclined  plane  (89,  h), 
F 


//  = 


-,    and   i^  =  /^  pf  cos  a. 


fFcos  oc 

The  coefficient  of  friction  is  a  constant  relation  in  a 
given  case,  depending  only  upon  the  nature  of  the  sur- 
faces in  contact. 

This  use  of  the  term  coefficient,  to  denote  a  constant  factor 
whose  value  depends  upon  the  substance  involved,  is  a  very  com- 
mon one  in  physical  science.  We  speak,  thus,  of  the  coefficient 
of  elasticity  of  a  certain  kind  of  steel;  the  coefficient  of  expan- 
sion, etc. 

92.  Angle  of  Friction.  The  tangent  of  the  angle  of 
friction  is  equal  to  the  coefficient  of  friction,  }jl  =  tan  a. 


Let  W  be  the  weight  of  a  body  (Fig.  38)  resting  on  a 
plane  HL,  and  suppose  that  the  plane  is  inclined  at  such 
an  angle  {a)  that  tlie  body  is  on  the  point  of  sliding. 


93.]  COEFFICIENT   OF   FRICTION.  97 

This  angle  is  called  the  angle  of  friction,  or  angle  of 
repose. 

The  force  of  friction  {F),  acting  in  the  direction  aF, 
must  be  equal  to  the  component  of  the  weight,  viz.  ad 
(=  IT  sin  a,  as  explained  in  Art.  89,  h),  which  urges  the 
body  down  the  plane;  that  is, 

F  =  W  sin  a,  (1) 

Also,  the  normal  pressure,  or  the  reaction  of  the  plane 
aR,  is  equal  to  «J,  the  portion,  of  the  weight  acting  per- 
pendicular to  LH;  but  ab  =  W  cos  a  (89,  b).     Hence 


Therefore 


and  since  ju 


R  =  Tfcosa.  (2) 

F         W^ina 

-—  =  -= =  tan  a; 

U  W  cos  a 

F_ 
R' 

,\  }x  =  tan  a. 


93.  Determination  of  the  Coefficient  of  Friction.     The 

preceding  article  gives  an  accurate  and  simple  method 
of  obtaining  the  value  of  the  coefficient  of  friction  by 
experiment.  If  the  surface  of  the  plane  is  made  of  one 
of  the  materials  in  question,  and  that  of  the  movable 
body  in  contact  with  it  of  the  other,  it  is  only  necessary 
to  observe  (Fig.  3S)  the  angle  {a  =  HLN)  at  which  the 
plane  must  be  elevated  in  order  to  put  the  body  on  the 
point  of  sh'din^;  then  (92)  tan  a  =  jjt. 

The  method  illustrated  by  Fig.  37  (p.  95)  may  also  be 
made  use  of.  For,  supposing  the  friction  of  the  pulley 
to  be  so  small  that  it  can  be  neglected,  if  the  weight  W, 
representing  one  of  the  surfaces,  is  known,  and  also  that 


yo  DYNAMICS.  [94. 

of  P  suflacient  to  put  W  on  the  point  of  moving  on  the 
other  required  surface,  then  the  constant  value  of  tlie 
ratio  of  these  quantities  is  equal  to  the  required  co- 

efficient  (^  =  ^  ^  ^^ 

94.  Examples  of    the  Coefficient  of  Friction.      The 

limiting  values  of  the  angle  of  friction  for  different 
groups  of  substances,  as  determined  by  experiment  in  the 
case  of  sliding  friction,  are  illustrated  by  Fig.  39. 


CO£COC 


Fig.  39. 

The  corresponding  range  in  the  values  of  the  coeffi- 
cient of  friction  is,  as  follows 

/t  (X  (Angle  of  Friction.) 

Bricks,  stones 0.60  -  0.73  31°  -  36° 

Wood  on  wood 0.19-0.47  11°  -  25° 

Metal  on  metal 0.14-0.22  8°  -  12^° 

Lubricants 0.05  -  0.11  3°  -    6° 

Many  different  circumstances  affect  these  values  ob- 
tained by  experiment,  so  that  the  above  are  only  to  be 
taken  as  average  results. 

For  rolling  friction  the  angle  is  much  smaller.  For  example,  it 
is  stated  that  a  railroad  train  in  good  order  on  a  good  road  is  not 


94.]  FEICTION.  99 

safe  against  starting  under  the  action  of  gravity  unless  the  gradi- 
ent is  less  than  18  to  20  feet  to  the  mile  (=  0°  13');  and  that,  if 
once  started,  the  train  will  continue  in  motion  on  gradients  as  low 
as  13  feet  per  mile  (Thurston). 

Ci)    ' 

EXAMPLES. 
XVI.  Friction.     Articles  82-94. 

1.  A  force  of  12  lbs.  is  just  sufficient  to  move  a  body  weighing 
48  lbs.  uniformly  along  a  horizontal  plane:  What  is  the  coeffi- 
cient of  friction? 

2.  The  value  of  yu  is  .3,  the  weight  of  the  body  is  16  lbs. :  What 
force  is  required  to  move  it  uniformly? 

3.  It  is  found  that  a  force  of  7  lbs.  suffices  to  move  a  body 
uniformly  on  a  horizontal  surface,  where  the  value  of  the  co- 
efficient of  friction  is  known  to  be  .25:  What  is  the  weight  of  the 
body? 

4.  A  body  weighing  15  lbs.  is  just  on  the  point  of  sliding  when 
the  surface  it  rests  upon  is  inclined  20°:  {a)  What  is  the  co- 
efficient of  friction  and  the  force  of  friction?  {h)  If  the  weight  of 
the  body  is  doubled,  what  values  have  these  quantities? 

5.  A  body  weighing  12  lbs.  rests  on  an  inclined  plane  whose 
angle  of  inclination  is  14°  and  where  //  =  .4:  What  is  the  force 
of  friction? 

6.  The  ratio  of  the  dimensions  of  an  inclined  plane  are  as  13 
(length)  to  5  (height)  to  12  (base):  Will  a  body  slide  If  the  co- 
efficient of  friction  is  {a)  .4  and  {h)  .5?  (c)  What  is  the  force  of 
friction  in  each  case,  the  weight  being  26  lbs.  ? 

7.  The  dimensions  of  the  plane  are  as  in  example  6 :  What  must 
be  the  value  of  the  coefficient  of  friction  if  the  force  of  friction  on 
the  plane  is  one  half  the  weight  of  the  body? 

8.  A  body  weighing  12  lbs.  rests  on  a  plane  where  the  co- 
efficient of  friction  is  .5:  What  is  the  force  of  friction  {a)  if  the 
plane  is  horizontal  ?  (5)  if  inclined  20°  to  the  horizon? 

9.  The  length,  height,  and  base  of  a  plane  are  30,  18,  and  24 
feet :  {a)  What  force  is  required  to  keep  a  body  weighing  20  lbs. 
from  sliding  down,  if  //  =  .2  ?  {p)  What  force  is  needed  to  draw 
it  uniformly  up  the  plane? 


100  DYNAMICS,^  — '      [94.^  ' 

^TA        •,.    ../  )••' 

10.  A  body  weighing  50  lbs.  rests  on  a  horizontal  plane,  where 
jii  =  .2i  What  force  is  required  to  move  the  bo^y, uniformly  if  it 
acts  as  a  pull  at  an  angle  of  30°  with  the  plane  (T'ig.  36)? 

11.  What  IS  the  force  required  to  move  the  body  in  example  10 
if  it  acts  at  the  same  angle  but  as  a  pus7i  (Fig.  35)  ? 

12.  A  force  of  60  lbs.  acting  as  a  ptdl  at  an  angle  of  20°  moves 
a  body  uniformly  on  a  horizontal  plane,  where  //  =  .3;  What  is 
the  weight  of  the  body? 

13.  If  the  conditions  in  example  12  are  fulfilled  when  the  force 
acts  at  the  same  angle  as  a  push,  what  is  the  weight  of  the  body? 

14.  A  body  weighmg  4  lbs.  is  held  against  a  rough  vertical  wall 
(//  =  .6)  by  a  force  acting  at  right  angles  to  the  wall:  What  is  the 
force? 

15.  A  force  of  120  lbs.  is  just  sufficient  to  support  a  body 
against  a  rough  vertical  wall  (//  =  .1);  the  force  acts  at  right 
angles  to  the  wall:  What  is  the  weight? 

[In  the  above  examples  no  distinction  is  made  between  the  re- 
sistance of  friction  when  the  body  is  just  on  the  point  of  moving, 
and  that  which  exists  when  the  body  is  already  in  uniform  motion  ; 
in  fact  the  former  is  sensibly  greater  than  the  latter.] 


/O    -     ^ 

CHAPTEE  v.— WOEK  AND  ENEEGY. 

A.    MECHANICAL    WORK — MEASUREMEl^-T    OF    WORK. 

95.  Definition  of  Work.  Work,  in  the  sense  in  which 
the  word  is  employed  in  Mechanics,  is  said  to  be  done 
when  a  force  acts  upon  a  body  and  motion  results  in 
the  direction  of  its  action. 

There  are  two  essential  elements  here:  (1)  the  force 
acting  to  overcome  a  resistance,  and  (2)  the  motion  pro- 
duced. Where  no  motion  results  from  the  action  of  a 
force,  no  work  is  done;  for  example,  a  column  support- 
ing a  weight  does  no  work. 

96.  Examples  of  Work.  Work  is  done  against  gravity 
when  a  man  lifts  a  weight  up  from  the  ground,  or  when 
he  ascends,  i.e.  lifts  himself  up,  a  hill;  against  friction 
when  a  horse  draws  a  carriage  along  a  horizontal  road; 
against  the  molecular  force  of  elasticity  when  a  spring  is 
wound  up  or  a  bow  is  stretched. 

97.  Measurement  of  Work.  The  ukit  of  work  is 
the  work  done  by  a  unit  force  acting  through  the  unit 
of  distance  one  foot.  If  the  force  be  expressed  in 
gravitation  measure  (72),  then  the  unit  of  work  is  the 
foot-jpound. 

(1)  The  work  done  by  a  constant  force  P,  acting 
through  a  distance  s,  is  equal  to  the  product  of  the 
force  into  the  distance;  that  is, 

the  work  done  by  P  =  P.s.  (1) 


iO^'  DYIS^AMICS.  [97. 

If  the  force  acts  obliquely,  at  an  angle  p  with  the 
direction  of  motion  (Fig.  40),  then  only  the  effective 
component  of  the  force,  P  cos  /?,  does  the  work.  The 
work  done  is,  therefore, 

P  cos  p.s,  (2) 

(2)  The  work  done  may  also  be  measured  by  the 
jp  effect  produced;  that  is,  when  a 

weight  W,  expressed  in  pounds, 
is  raised  through  a  height  h,  ex- 
^!^  pressed  in  feet,  the  work  done  is 


^®*  ^*  equal  to  the  product  of  the  weight 

raised  into  the  vertical  height ;  that  is, 

the  work  done  in  raising  a  weight  =  W.h.      (3) 

The  effective  distance  only — that  is,  the  vertical  height — 
is  considered.  If  one  pound  is  raised  vertically  one 
foot,  then  one  foot-pound  of  work  is  done,  and  this  is 
the  simplest  form  of  the  unit  of  work  as  above  defined. 

If  the  weight  is  an  extended  body,  or  if  a  number  of  bodies 
are  considered  together,  the  height  to  be  talten  is  the  vertical  dis- 
tance through  which  the  centre  of  gravity  (defined  in  Art.  159)  is 
raised. 

(3)  Still,  again,  if  a  uniform  resistance  {R)  is  over- 
come through  an  effective  distance  d,  then  the  work 
done  is  equal  to  the  product  of  the  resistance  into  the 
distance;  that  is, 

the  work  done  against  a  resistance  =  R.d.      (4) 

It  is,  in  fact,  immaterial,  in  the  estimation  of  the 
amount  of  work  done  in  any  case,  whether  the  attention 
be  directed  to  the  force  acting,  on  the  one  hand,  or  the 
weight  raised,  or  resistance  overcome,  on  the  other.    For 


99.]  MEASUREMENT   OF  WOEK.  103 

in  all  cases  it  follows  from  the  law  of  the  Conservation 
of  Energy,  as  explained  later  (101  et  seq.),  that  the  two 
must  be  equal;  that  is, 

P.s^W.h,  (5) 

or 

P.s  =  R.d. 

Whenever  a  weight  is  raised,  the  simplest  method  of 
estimating  the  amount  of  work  done  is  by  the  product 
W.h»  In  some  cases,  however,  it  is  more  simple  to 
measure  the  force  acting  and  the  distance  through 
which  it  acts,  and  to  obtain  the  number  of  foot-pounds 
of  work  done  from  the  product  P.s» 

98.  When  the  distance  through  which  P  acts  and  W  is 
raised  differ,  as  when,  for  example,  by  means  of  a  set  of  pul- 
leys a  small  power  acting  through  a  great  distance  raises  a 
large  weight  through  a  small  height,  the  equation  above 

p 
(5)  shows  that  the  ratio  of  -r^  is  the  inverse  ratio  of  the 

distances  through  which  they  act,  or  -tiF  =  — >  ^^^^  ^^ — 

VY  S 

The  Power  is  to  the  Weight  as  the  height  through  which 
the  Weight  is  raised  is  to  the  distance  through  which  the 
Poiver  acts.  This  principle  will  be  employed  in  the 
discussion  of  the  various  mechanical  contrivances  or 
machines,  the  lever,  wheel  and  axle,  etc.,  in  Chapter 
VIII. 

99.  Rate  of  Work.  The  rate  of  work  is  measured  by 
the  amount  of  work  done,  for  example  by  a  steam- 
engine,  in  a  unit  of  time.  The  ordinary  unit  employed 
is  called  the  horse-power,  and  is  equal  to  550  foot- 
pounds per  second,  or  33,000  foot-pounds  per  minute. 
This  unit  is  employed  in  measuring  the  efficiency 
of  a  steam-engine. 


104 


DYNAMICS. 


[100. 


100.  Applications  of  the  above  Principles.  Suppose  a 
constant  force  (P)  acts  to  draw  a  body  along  a  rough  hori- 
zontal plane  for  a  distance  s.  The  work  done  is  then 
equal  to  F  X  s  [97  (1)].  But  as  it  is  difficult  to  determine 
the  value  of  P  directly,  the  principle  may  be  made  use  of 
that:  If  the  body  moves  uniformly,  this  force  must  be 
just  equal  to  the  force  of  friction;  that  is,  P  =  F,  But 
if  the  weight  of  the  body  is  W  and  the  coefficient  of 
friction  ^  is  known,  then  F=  }xR  =  /^Tf  (91),  and 

the  work  done  against  friction  =  /xW,s, 

Suppose  a  weight  W  (Fig.  41)  is  raised  to  the  top  of 


Fio41. 

an  inclined  plane  whose  height  is  h,  and  whose  angle  of 
inclination  HLK  is  a-,  then  (97) 

the  work  done  in  raising  the  weight  =  W.h, 

Also,  if  I  is  the  length  of  the  plane  and  //  the  coefficient 
of  friction  for  the  surfaces  in  contact,  since  F=  }xR  = 
^Wcosa  (89,  91), 

the  work  done  against  friction  =  F.l  =  piW coa  a.l, 

and  the  total  amount  of  work  done  is  equal  to 

Wh  -\-  /^W 008  a.l 


100.]  WORK.  105 

Since  h  =  I  sin  a,  this  may  be  put  in  the  form 

Wl  (sin  a-f  //  cos  a); 

.       .  h 

or  again,  since  cos  a  =  j, 

Wh-\-  fxWb, 

The  last  expression  shows  that  the  work  done  in  the 
case  supposed  is  the  same  as  that  required  to  drag  the 
body  from  L  to  K,  and  then  to  raise  it  vertically  to  H. 

EXAMPLES. 
XVII.   Work.    Articles  95-100. 

1.  A  weight  of  600  lbs.  is  raised  to  the  top  of  an  inclined  piano 
whose  length  is  1200  feet,  and  the  angle  of  inclination  =  10° : 
What  work  is  done? 

2.  A  weight  of  150  lbs.  is  raised  to  the  top  of  a  tower  along  a 
spiral  path  half  a  mile  long,  and  which  winds  about  it  rising  at  a 
uniform  angle  of  15° :  How  much  work  is  done  on  the  weight? 

3.  If  a  man  in  walking  raises  his  centre  of  gravity  a  distance 
equal  to  ^  of  the  length  of  his  step  (as  has  been  estimated),  how 
much  work  will  he  do,  if  he  weighs  150  lbs.,  in  walking  20  miles? 

4.  How  much  work  is  done  by  an  engine  weighing  20  tons  in 
running  a  distance  of  a  mile  on  a  horizontal  track,  if  the  total  rC' 
sistance  is  120  lbs.  per  ton? 

5.  A  sled  weighing  1500  lbs.  is  dragged  10  miles  on  the  snow, 
where  the  coefficient  of  friction  is  .075:  What  work  is  done 
against  friction? 

6.  How  much  work  is  done  against  friction  in  dragging  a 
weight  of  400  lbs.  a  distance  of  1000  yards  along  a  horizontal 
plane,  if  the  coefficient  of  friction  is  .5? 

7.  A  weight  of  250  lbs.  is  dragged  up  an  inclined  plane  whose 
length  is  2600  feet  and  the  height  is  1000  feet  (/^  =  .3) :  How  much 
work  is  done? 

8.  A  weight  of  130  lbs.  is  drawn  along  a  horizontal  plane  for  a 


106  DYNAMICS.  [101, 

distance  of  1000  feet,  and  then  up  an  inclined  plane  {a  =  30°)  for 
tlie  same  distance;  the  coeflSicient  of  friction  is  .3  for  both  sur- 
faces: What  worlt  is  done? 


B.  ENERGY — COKSERVATION"  AND  CORRELATION  OF 
ENERGY. 

101.  Definition  of  Energy.  Energy  is  the  capacity  of 
performing  work. 

The  first  grand  principle  or  doctrine  of  energy  is  called 
the  Conservation  of  Energy,  which  states  that : 

The  various  forms  of  energy  may  he  changed  into  one 
another,  hut  the  sum  total  remains  the  same;  no  energy 
is  ever  lost.  This  is  a  fundamental  principle  in  all 
physical  science,  and  its  importance  cannot  be  overesti- 
mated. It  is,  in  a  certain  sense,  a  corollary  from,  and  an 
extension  of,  the  third  law  of  motion  (69).  Like  these 
laws  its  truth  has  been  established  by  extended  series  of 
observations  and  physical  experiments  (66). 

As  a  proper  understanding  of  this  subject  is  neces- 
sary for  a  thorough  comprehension  of  the  laws  of  Me- 
chanics, and  at  the  same  time  as  this  is  impossible 
without  a  somewhat  extended  discussion  of  the  subject, 
it  is  necessary  at  the  outset  to  treat  it  broadly  in  its  ap- 
plication to  all  Physics  and  not  only  as  restricted  to 
Mechanics. 

The  term  worh  must,  in  the  first  place,  be  understood 
as  having  a  wider  signification  than  that  given  in  the 
preceding  portion  of  this  chapter.  Taken  broadly,  work 
is  involved  in  the  production  of  any  physical  or  chemical 
change.  For  example,  work  is  done  not  only  when  a 
mass  of  iron  is  raised  from  the  ground,  but  also  when  by 
heat  its  temperature  is  raised;  so,  too,  when  water  is 
changed  from  its  liquid  form  into  steam ;  when  an  elec- 


103.]         KINETIC   AND  POTENTIAL   ENERGY.  107 

trical  current  is  sent  through  a  wire,  as  in  the  telegraph; 
and  when  the  atoms  of  carbon  of  the  coal  unite  with 
the  oxygen  of  the  air  and  combustion  ensues  accom- 
panied by  heat  and  light. 

102.  Forms  of  Energy.  The  forms  of  energy  are 
divided  into  the  following  classes: 

(1)  Mechanical  energy,  or  the  visible  energy  of 
masses  of  matter,  including  the  energy  due  to  elasticity. 

(2)  Molecular  energy,  including  heat,  light,  electricity, 
and  magnetism.  " 

(3)  Chemical  energy,  or  that  produced  by  the  chemical 
union  of  unlike  atoms. 

The  forms  of  energy  are  also  divided  into  {a)  Kinetic  * 
energy,  or  energy  of  motion,  and  {h)  Pote7itial  energy, 
or  energy  of  position.  It  will  be  sufficient  here  to  apply 
this  distinction  more  particularly  to  mechanical  energy, 
but  it  also  belongs  to  the  forms  which  come  under  the 
other  heads. 

103.  Kinetic  and  Potential  Energy.  (1)  Kii^etic 
ENERGY,  mechanically  considered,  is  that  which  belongs 
to  bodies  in  virtue  of  their  motion.  Every  moving  body 
has  a  certain  amount  of  energy,  or  capacity  of  perform- 
ing work,  in  consequence  of  this  motion.  For  example, 
a  swiftly  moving  cannon-ball,  a  running  stream,  the  wind 
— all  these,  because  of  their  motion,  have  a  definite  power 
of  doing  work.  In  fact,  their  motion  is  a  result  of  energy 
expended  upon  them  at  some  previous  time,  which  they 
will  themselves  give  back  if  their  motion  is  arrested. 
This  energy  of  motion  is  sometimes  called  vis  viva,  or 
**  living  force,"  and  sometimes  accumulated  work. 

(2)  Potential  energy  is  that  which  belongs  to  bodies 

*  From  the  Greek  word  xireoo,  to  move. 


108  DYNAMICS.  [101 

in  virtue  of  their  position.  Every  body  situated  above 
the  surface  of  the  earth  has  a  tendency  to  fall  to  it, 
under  the  action  of  gravity,  and  this  determines  its 
potential  energy,  or  energy  of  position.  For  example, 
the  weights  of  a  clock,  when  wound  up,  have,  because  of 
their  elevated  position,  a  power  of  doing  work ;  e.g.^  in 
turning  the  machinery.  So,  too,  a  body  of  water 
behind  a  mill-dam  represents  a  certain  amount  of  poten- 
tial energy;  that  is,  of  energy  which  may  be  expended, 
e.g.  in  driving  a  mill  if  the  water  be  allowed  to  fall. 
Also,  the  stones  and  bricks  in  a  building  represent  a 
certain  amount  of  energy  expended  once  in  raising  them, 
and  now  present  only  potentially  because  of  their  posi- 
tion. A  watch-spring  when  wound  up,  and  a  bow  when 
stretched,  are  other  examples  of  potential  energy. 

In  every  such  example  the  position  of  the  body  in 
question  (or  of  its  molecules),  as  the  velocity  in  the  pre- 
ceding case,  indicates  that  a  certain  amount  of  energy 
has  been  expended  upon  it,  and  this,  as  before  stated, 
will  be  given  back  on  a  return  to  the  original  position. 

The  level  of  the  sea  may  be  made  the  surface  of  refer- 
ence for  convenience;  strictly,  a  terrestrial  body  would 
have  potential  energy  anywhere  except  exactly  at  the 
centre  of  gravity  of  the  earth.  Moreover,  other  levels 
may  be  taken  as  the  zero;  for  example,  the  weights  of  a 
clock,  just  mentioned,  may  be  said  to  have  no  potential 
energy  when  they  have  fallen  to  the  lowest  point  in 
their  course. 

104.  Measurement  of  Energy,  {a)  Inasmuch  as  the 
amount  of  work  done — that  is,  in  other  words,  energy 
expended — in  raising  a  weight  W  through  a  vertical 
height  h  is  equal  to  Wh,  it  is  evident  that: 


105.]         KINETIC   AND  POTENTIAL  ENEEGT.  109 

The  POTENTIAL  ENERGY  of  any  body  is  equal  to  the 
product  of  its  iveight  into  the  distance  it  has  to  fall. 

(b)  Again:  since,  if  a  body  of  weight  TTfall  through 
a  height  h,  its  energy  of  position  is  entirely  changed 
into  energy  of  motion,  this  last  or  kinetic  energy  will  be 
equal  to  Wh.  But  in  falling  freely  from  rest  through  a 
distance  h,  a  body  acquires  a  velocity  (27)  such  that 

v'  =  2ghy  and  h  =  ^.    Substituting  this  value  of  h,  the 

kinetic  energy  becomes  -«— •    But  the  kinetic  energy  of 

any  body  of  weight  W  and  velocity  v  must  be  the  same 
as  that  of  the  body  which  has  acquired  this  velocity  in 
falling.  Hence,  in  general,  the  energy  of  a  body  in  mo- 
tion is  expressed  by 

^' 

W 
Since  the  mass  M=  —  (68),  the  value  above  may 

be  expressed  in  this  form: 
M    , 

The  KINETIC  ENERGY  of  any  body  is  equal  to  the  pro- 
duct of  one  half  the  mass  into  the  square  of  the  velocity, 

105.  A  body  of  weight  W  and  moving  with  a  velocity 
V  will,  if  brought  to  rest,  expend  against  the  resistance 

an  amount  of  work  equal  to  -^r— .     If  the  resistance  is 

uniform  (as  is  true  of  friction),  then,  by  the  principles 
explained  in  Art.  97  (3),  we  shall  have 


110  DYI^AMICS.  [lOa 

Here  -7; —  is  the  amount  of  work  accumulated,  or 
2g 

stored  up,  in  the  moving  body,  and  B.  d  is  the  work  done 
against  the  resistance.  If  B  is  known,  then  d  can  be 
calculated,  and  vice  versa.  In  the  case  of  friction  on  a 
horizontal  plane,  for  example,  B  =  F  =  /xW  (91),  and 
hence  the  distance  a  given  body  will  slide  can  be  easily 
computed. 

When  d  is  very  small,  as  when  a  heavy  weight  descend- 
ing drives  a  pile  a  short  distance  in  the  mud,  then  B,  the 
average  resistance^  will  be  obviously  very  great.  It  is  on 
this  principle  that,  when  a  very  great  resistance  has  to  be 
overcome,  it  is  often  most  effectual  to  make  use  of  the 
large  amount  of  energy  stored  up  in  a  moving  body  of 
considerable  mass,  which  may  be  expended  through  a 
very  small  distance  ;  consider  the  efficiency  of  a  heavy 
hammer. 

106.  Eelation  of  Kinetic  Energy  to  Momentum.     The 

distinction  must  be  carefully  made  between  the  momen- 
tum and  the  kinetic  energy  of  a  moving  body.  Suppose 
the  mass  of  the  body  to  be  M,  and  the  velocity  v  ;  then 

the  momentum  =  M.v, 

M 

the  kinetic  energy  =  -^.v'. 

From  these  values  it  is  seen  that  for  the  same  body  the 
momentum  is  proportional  to  the  velocity,  but  the  ki- 
netic energy — that  is,  the  power  of  doing  work — to  the 
square  of  the  velocity.  For  example,  suppose  two  bodies 
to  have  the  same  mass,  but  let  the  velocity  of  one  be  160 
feet  per  second,  while  that  of  the  other  is  80  feet  per 
second;  the  former  will  have  tivice  the  momentum,  but 


107.]  KINETIC  AND  POTENTIAL   ENERGY.  Ill 

its  kinetic  energy  will  be  four  times  greater.  In  other 
words,  the  first  body  can  do  four  times  as  much  work  in 
coming  to  rest;  it  will  ascend  vertically  upward  against 
gravity  four  times  as  far;  that  is,  400  feet  instead  of  100 
feet.  So,  too,  it  will  penetrate  four  times  farther  into 
a  log  of  wood  swung  as  that  described  in  Art.  70  as  a 
ballistic  pendulum,  tliough  the  momentum  given  to  the 
moving  mass  will  be  only  twice  as  great. 
In  the  relation  in  Art.  70,  viz., 

MV  ±  mv  =  {M  -\-  m)  v', 

it  was  shown  that  the  momentum  of  the  two  bodies 
after  impact  was  equal  to  the  sum  of  their  momenta 
taken  separately  before,  they  being  supposed  to  be  per- 
fectly inelastic.  This  is  not  true  of  the  kinetic  energy. 
Suppose  the  weights  of  two  bodies  to  be  96  and  64  lbs. ; 
then  Jlf^:  3  (=  Jf )  and  m  =  2  (=  f|):  also,  let  V=  100 
feet  per  second,  and  z;  =  50;  then,  by  the  above  equa- 
tion, v'  =  80.  Now  the  sum  of  the  kinetic  energies  of 
if,  (iMV)  SLndm,{i7nv')  is  equal  to  17,500  ft.lbs.,  but 
the  kinetic  energy  of  them  moving  together  with  the 
new  velocity  v'  is  only  16,000  ft.lbs.  This  difference 
would  be  still  greater  if  the  original  motions  were  in 
opposite  directions;  viz.,  17,500  ft.lbs.  and  4000  ft.lbs, 
respectively. 

Hence  there  is  an  apparent  loss  of  energy  after  impact. 
The  equivalent  of  this  loss  is  to  be  found  in  the  heat 
produced  by  the  blow,  as  explained  further  in  articles 
108  et  seq.  This  relation  must  be  brought  in  in  order 
to  make  Newton's  third  law  of  motion,  in  its  general 
application,  rigidly  true. 

107.  Transformation  of  Kinetic  and  Potential  Energy. 
The  two  kinds  of  energy,  considered  in  articles  103  and 


112 


DYNAMICS. 


[107. 


104,  may  be  transformed  the  one  into  the  other,  and,  if 
no  other  form  of  energy  appears,  the  law  of  the  Coi^- 
SERYATIOK  OF  Ei^TERGY  requires  that  this  interchange 
should  go  on  without  loss.  For  example,  suppose  a 
ball  (Fig.  42)  is  rigidly  attached  by  a  rod  to  the  sup- 
port C,  about  which  it  turns  without  friction.     If  its 


Fig.  43. 

position  be  changed  from  ^  to  ^,  and  it  be  supported 
here  for  a  moment,  it  is  evident  that  a  certain  amount 
of  work  has  been  performed,  or  energy  expended,  upon 
it  represented  exactly  by  W.7i,  where  7i  is  the  vertical 
height  DB.  This  amount  of  energy  has  been  imparted 
to  the  body,  and  belongs  to  it  as  potential  energy  in  vir- 
tue of  its  position. 

Now  suppose  the  ball  to  descend;  at  each  successive 
point  as  J/,  in  its  course  from  A  back  to  B,  it  loses  part 
of  its  potential  energy,  but  gains  a  corresponding  amount 
of  kinetic  energy,  or  energy  of  motion;  when  B  is  reached, 

/  Wv''  \ 

all  its  energy  is  that  of  motion  I  Wh  =  -z — J.    In  virtue 

of  this  motion  it  will  ascend  on  the  other  side,  exchang- 
ing at  each  point  its  energy  of  motion  for  energy  of 
position,  and  at  A'  its  velocity  is  0  and  its  energy  all 
potential. 


107.]         KINETIC  AND  POTENTIAL  ENEEGY.  113 

Again,  on  descending,  the  excliange  of  energy  takes 
place  as  before.  If  the  supposed  conditions  of  a  per- 
fectly rigid  rod,  moving  without  friction  at  C  and  meet- 
ing with  no  resistance  of  the  air,  could  be  realized,  this 
motion  would  go  on  forever;  it  would  be  one  kind  of 
perpetual  motion.  [It  would  not  be  the  '*  perpetual 
motion"  sought  for  ;  for  example,  an  engine  which  shall 
go  on  doing  work  forever  without  being  supplied  with 
fuel;  this  the  doctrine  of  energy  shows  to  be  absurd  and 
impossible.] 

Another  example  will  further  illustrate  the  transformation  of 
the  two  forms  of  energy.  Suppose  a  ball  to  be  projected  from 
the  ground  vertically  upward,  with  a  velocity  v;  the  energy  at 
the  moment  of  starting  is  all  kinetic.  By  the  laws  of  kinematics 
(44)  it  will,  if  gravity  alone  resists  it,  ascend  to  a  height  (Ji)  so 

that  h  =^  -jr--.    As  it  ascends  its  kinetic  energy  is  continually  ex- 

changed  for  energy  of  position,  and  at  its  highest  point  it  has  only 
potential  energy.    If  its  weight  is  W,  the  amount  of  work  done  is 

Wh,  but  h  ■=  —-\  hence  the  work  done,  which  is  the  equivalent 

2g 

of  the  initial  kinetic  energy,  is  -^r—  (same  result  reached  as  in  Art. 

104,  b).  Further,  on  its  descent  it  will  continually  exchange  its 
energy  of  position  for  that  of  motion,  and  when  it  reaches  the 
ground  its  energy  is  all  kinetic.  Suppose  that  the  ball  and  the 
surface  it  comes  in  contact  with  are  perfectly  elastic.  The  result 
of  the  blow  will  be  to  compress  the  molecules  of  the  body  for  an 
instant,  and  at  this  moment  it  is  at  rest  and  its  energy  is  repre- 
sented potentially  by  the  new  position  of  the  molecules.  In  con- 
sequence of  the  elasticity,  however,  the  molecules  tend  to  regain 
their  original  position,  and  thus  the  potential  energy  is  again 
transformed  into  kinetic,  and  the  effect  is  to  project  the  ball  up- 
ward with  the  same  velocity  as  before.  In  the  ideal  case  supposed 
this  exchange  would  go  on  continually,  and  the  ball  would  bound 
forever. 


114  DYNAMICS.  [108. 

108.  Apparent  loss  of  Visible  Energy.  It  is  obvious 
that  the  conditions  supposed  in  the  preceding  article 
cannot  be  realized,  but  that  the  pendulumx  and  the 
bounding  ball  will  sooner  or  later  come  to  rest.  In  such 
cases  there  is  an  appare7it  loss  of  visible  energy.  Still 
more  is  there  an  apparent  loss  of  energy  when  the 
motion  of  a  train  is  arrested  at  a  station,  or  that  of  a 
cannon-ball  by  a  target.  In  such  cases  it  was  once  be- 
lieved that  the  energy  was  really  lost,  but  it  is  now 
known  that  this  is  as  untrue  as  it  would  be  to  suppose 
that  the  matter  in  a  piece  of  paper  is  lost  when  it  is 
burned.  The  matter  is  unchanged  in  amount,  and  is 
truly  indestructible,  though  its  form  may  be  altered  and 
it  so  become  invisible  to  the  eye.  In  an  analogous  way, 
in  this  apparent  loss  of  visible  energy,  a  new  form  of 
energy  takes  the  place  of  that  which  disappears,  for  the 
energy  itself  is  indestructible.  This  form  of  energy, 
produced  as  the  equivalent  of  the  mechanical  energy 
which  has  disappeared,  is  heat, 

109.  Nature  of  Heat.  Heat  is  now  believed  to  be, 
not  a  form  of  matter,  as  once  supposed,  but  a  "mode 
of  motion;"  more  particularly  it  is  a  very  rapid  un- 
dulatory  vibration  of  the  particles  of  matter  making  up 
the  heated  body.  When  heat  is  transmitted  through  a 
medium  without  raising  its  temperature  it  is  said  to  be 
radiated,  and  the  undulatory  motion  is  believed  to  be 
propagated  at  a  very  great  velocity  by  the  particles  of  a 
supposed  elastic  fluid  called  the  ether.  Thus,  the  heat 
of  a  stove  is  said  to  be  radiated  in  all  directions  from  it; 
so,  too,  the  heat  of  the  sun  is  said  to  be  radiated  to  the 
earth,  and  the  heat  received  is  called  radiant  heat,  or 
radiant  energy. 


111.]  MECHANICAL  ENEEGY  AND  HEAT.  115 

When,  however,  the  heat  is  transmitted  through  a 
body  at  a  comparatively  slow  rate,  as  from  one  end  of 
an  iron  rod  thrust  in  a  furnace  to  the  other,  it  is  said  to 
be  conducted,  and  in  this  case  the  particles  of  the  bar 
itself  are  believed  to  propagate  the  motion. 

A  hot  body  is  one  whose  particles  are  in  rapid  motion; 
but  *'hot,"  as  the  word  is  used,  is  only  a  relative  term, 
for  this  motion  belongs  to  the  molecules  of  all  bodies  of 
which  we  have  any  knowledge,  however  **  cold,"  and  the 
rapidity  of  the  motion  determines  the  degree  of  heat 
(temperature)  as  manifested,  for  example,  to  our  senses 
or  to  a  thermometer. 

110.  Examples  of  the  Production  of  Heat  from  Me- 
chanical Energy.  Examples  of  the  appearance  of  heat 
at  the  same  time  with  the  disappearance  of  visible 
energy  are  very  common.  Thus,  a  nail  rubbed  quickly 
with  a  file  becomes  **  hot;"  the  same  is  true  of  a  metallic 
button  rubbed  on  a  piece  of  cloth.  A  piece  of  iron  on 
an  anvil  may  be  raised  to  a  dull  red  heat  by  rapid  blows 
from  a  hammer;  a  friction-match  is  ignited  by  the  heat 
produced  by  a  scratch;  a  cannon-ball  whose  motion  is 
arrested  by  a  target  is  itself,  as  well  as  the  target,  very 
much  heated  by  the  collision.  In  all  such  cases,  as  will 
be  apparent  from  what  has  been  said,  the  visible  mass 
energy  is  exchanged  for  the  molecular  energy  of  heat; 
the  slow  motion  of  the  mass  for  the  very  rapid  motion 
of  the  molecules. 

111.  Definite  Relation  between  Heat  and  Mechanical 
Energy.  If  it  be  true  that  the  heat  produced  in  the  cases 
named  (110)  is  the  equivalent  of  the  visible  energy  lost, 
it  follows  that  there  must  be  a  definite  numerical  ratio 
between  a  certain  amount  of  work  done  and  the  amount 


116  DYNAMICS.  [111. 

of  heat  produced  by  it.  This  relation  has  been  estab- 
lished in  many  different  ways  by  different  experimenters, 
but  in  all  the  cases  the  essential  part  of  the  process  is 
the  same,  viz.,  to  measure  the  amount  of  mechanical 
energy  expended  (in  foot-pounds),  and  also  to  determine 
the  amount  of  heat  produced  as  its  equivalent. 

Heat  is  measured  in  heat-units;  that  is,  the  unit  of  heat  is 
y  d  that  amount  of  heat  required  to  raise  one  pound  of 
water  one  degree  in  temperature.  For  physical  prob- 
lems the  Centigrade  thermometer  is  universally  em- 
ployed ;  but  with  English-speaking  people  the  Fahren- 
heit thermometer  is  commonly  used  as  the  house 
thermometer.  The  relation  of  the  two  is  evident  from 
Fig.  43.  For  the  Centigrade  thermometer  the  freezing- 
point  of  water  is  made  the  zero,  and  the  distance  from 
it  to  the  boiling-point  is  divided  into  100  degrees.  In 
the  Fahrenheit  thermometer  the  freezing-point  is  32  de- 
grees above  the  zero,  and  the  boiling-point  212  degrees 
above.  Hence  100  degrees  Centigrade  correspond  to 
180°  (=  212°  -  32°)  Fahrenheit.  To  change  the  read- 
Fio.  43.      jjjgg  Qf  either  thermometer  to  those  of  the  other,  we 

have  representing  the  number  of  degrees  in  the  two  scales  by  F 

and  (7  respectively, 

fC-f  32°  =  F,        and       ^{F-  32°)  =  G, 

The  method  which  was  employed  by  Joule*  was  es- 
sentially as  follows:  A  metal  box  B  (Fig.  44)  was  taken 
full  of  water;  in  this  was  placed  a  paddle  (Fig.  45)  at- 
tached to  a  vertical  spindle  A,  which  could  be  revolved 
by  means  of  a  string  passing  over  two  pulleys  G  and  B, 
and  attached  to  two  known  weights  E  and  F,     If  now 

*  The  experiments  of  Dr.  Joule  were  carried  on  between  1843 
and  1849.  He  employed  several  different  methods  for  determin- 
ing the  '  *  mechanical  equivalent  of  heat, "  but  that  which  led  to 
the  most  satisfactory  results  is  the  one  here  described. 


11 


111.] 


MECHANICAL  ENERGY  AND  HEAT. 


117 


these  weights  {E  -\-  F  =  W)  are  allowed  to  descend 
freely  through  a  distance  li,  marked  on  the  vertical 
scales,  the  work  done  is  Wh\  but  this  is  expended  in 
turning  the  paddle  in  the  water,  and,  owing  to  the  fric- 
tion of  the  water  (Art.  87),  is  all  transformed  into  heat. 
Hence  if  the  amount  of  water  is  known,  and  its  tempera- 
ture before  and  after  the  experiment,  the  number  of 
foot-pounds  of  work  required  to  produce  one  heat-unit — 
that  is,  to  raise  1  lb.  of  water  1°  C. — can  be  readily- 
calculated.  In  the  actual  experiment  it  was  necessary 
to  make  corrections  for  the  loss  of  energy  in  the  friction 


Fio.  44. 


Fig.  45. 


of  the  pulleys,  the  radiation  of  heat  from  the  box,  and 
several  other  points  which  need  not  be  explained  here. 
The  result  obtained  was  this :  That  an  expenditure  of 
1390  foot-pounds  of  work  produce  one  unit  of  heat  on  the 
Centigrade  scale  (772  ft.  lbs.  on  the  Fahrenheit  scale). 
Many  other  experiments  have  been  made  in  various  ways 
having  as  their  object  the  determination  of  this  same 
relation;  for  example,  the  amount  of  heat  produced  by 
the  friction  of  two  iron  plates  in  mercury,  that  caused 
by  the  collision  of  two  heavy  bodies  one  of  which  has 


118  DYNAMICS.  [112. 

fallen  through  a  known  height,  and  so  on.  All  these 
experiments  have  confirmed  the  relation  obtained  by 
Joule. 

112.  Conversion  of  Heat  into  Work.  Since  a  definite 
amount  of  mechanical  work  is  equivalent  to  a  certain 
amount  of  heat  energy,  the  converse  must  also  be  true: 
that  heat  is  convertible  into  mechanical  work.  This 
conversion  of  the  former  kind  of  energy  into  the  other 
is  best  seen  in  the  steam-engine.  Here  the  burning  coal 
in  the  furnace  converts  the  water  of  the  boiler  into 
steam,  and  the  expansion  of  this  steam  in  the  steam- 
chest  connected  with  the  condenser  sets  the  piston  in 
motion  backwards  and  forwards.  This  is  mechanical 
motion,  and  it  may  be  utilized,  for  example,  to  drive 
the  lathes  in  a  machine-shop,  to  pump  up  water,  to 
propel  a  steamship  or  a  train  of  cars.  All  these  are 
cases  in  which  from  heat  mechanical  work  is  obtained. 

There  is  one  most  important  difference  between  the 
two  cases  that  have  been  described;  viz.,  the  transforma- 
tion of  work  into  heat,  and  that  of  heat  into  work.  The 
former  transformation  can  be  completely  made,  but 
under  no  conditions,  which  are  practicably  obtainable, 
can  a  certain  amount  of  heat  be  all  changed  into  me- 
chanical work.  A  perfect  **  reversible  engine"  (as  that 
of  Carnot)  can  be.  conceived  of,  but  the  necessary  con- 
ditions cannot  in  practice  be  even  approximately  realized. 

113.  Other  Forms  of  Molecular  Energy.  The  form 
of  molecular  energy  most  closely  related  to  Mechanics 
is  that  just  considered;  viz.,  heat.  There  are,  how- 
ever, other  forms  of  energy  into  which  mechanical  work 
may  be  converted,  and  conversely  from  which  work  may 
be  obtained.     Here  belong  the  other  physical  agents. 


114.]  TEANSFOKMATION   OF  ENEKGY.  119 

light,  electricity,  magnetism.  The  full  understanding 
of  their  mutual  relations  requires  an  extended  knowl- 
edge of  Physics,  and  cannot  be  attempted  here,  but  some 
illustrations  will  suffice  to  make  the  subject  clear. 

114.  Examples  of  Transformation  of  Energy.     The 

production  of  work  from  burning  coal,  already  given,  is 
one  example.  Another  example  is  this:  The  water  in 
a  mill-pond  represents  a  certain  amount  of  potential 
energy  (103);  this  in  falling  through  the  mill-race  may 
be  made  to  turn  a  mill-wheel,  and  its  motion  represents 
a  certain  amount  of  mechanical  energy  derived  from  the 
water;  if  a  turbine  wheel  is  employed,  from  60  to  80  per 
cent  of  the  energy  of  the  water  may  be  utilized  under 
favorable  conditions.  The  motion  of  the  wheel  may  be 
given  to  a  saw,  which  shall  do  work  in  overcoming  the 
cohesion  of  the  wood,  or  to  millstones,  by  which  grain 
is  ground.  Or,  again,  it  may  turn  an  electro-magnetic 
machine.  This  cannot  be  described  here,  but  it  is  suffi- 
cient to  understand  that  the  machine  accomplishes  this: 
that  the  rapid  mechanical  motion  results  in  the  produc- 
tion of  a  current  of  electricity,  which  is  the  equivalent 
of  a  certain  portion  of  the  mechanical  energy.  This 
electrical  energy  may  be  conveyed  by  a  copper  wire  for  a 
considerable  distance  (with  a  loss,  however,  for  some  of 
it  is  inevitably  transformed  into  useless  heat),  and  then 
the  electricity  may  be  used  to  make  a  light,  when  the 
remainder  of  the  energy  is  transformed  into  light  and 
heat,  or  it  may  be  used  to  do  work  in  chemical  separa- 
tion, as  in  electro-plating  with  copper.  Still,  again,  it 
may  by  means  of  a  second  similar  machine  be  trans- 
formed back  again  into  mechanical  motion,  and  this 
used  to  do  any  kind  of  work  desired. 

Again,  the  revolution  of  a  disc  of  plate-glass  between 


120  DYNAMICS.  [115. 

cushions  suitably  arranged  may  be  made  to  produce 
electricity,  which,  is  the  equivalent  of  part  of  the  me- 
chanical energy  expended  in  turning  the  wheel,  the 
remainder  being  expended  against  friction  and  resulting 
in  heat.  If  this  electricity  is  collected  on  a  brass  cylin- 
der and  then  a  spark  taken  from  it,  the  light  and  heat  of 
the  spark,  with  the  vibratory  motion  resulting  in  the 
noise,  are  the  forms  of  energy  into  which  the  mechanical 
work  has  been  transformed. 

Chemical  affinity  also  represents  a  most  important 
kind  of  energy.  Two  dissimilar  atoms  under  suitable 
conditions  combine,  and  some  other  form  of  energy  is 
the  result.  Thus,  in  the  combustion  of  coal  in  air,  it  is 
the  union  of  the  carbon  and  hydrogen  of  the  coal  with 
the  oxygen  of  the  air  which  results  in  the  formation  of 
heat  energy.  So,  too,  the  charcoal,  nitre,  and  sulphur 
mixed  together  in  gunpowder  will  unite  under  proper 
conditions,  and  the  result  is  the  appearance  of  energy 
which  produces  not  only  heat  and  liglit  (and  noise),  but 
also  may  do  a  vast  amount  of  mechanical  work. 

116.  Conservation  of  Energy.  To  all  the  examples  of 
the  transformation  of  one  form  of  energy  into  another, 
given  in  the  preceding  article,  the  law  of  the  Conserva- 
tion of  Energy  applies.  It  requires  that  the  sum  total 
should  remain  the  same,  that  no  energy  should  be  lost 
in  the  changes.  In  order  to  prove  this  rigidly  it  would 
be  necessary  to  be  able  to  correlate  all  the  different 
kinds  of  energy  and  express  between  them  a  definite 
ratio,  as  that  between  heat  and  mechanical  energy.  This 
cannot  always  be  done,  for  of  the  real  nature  of  some  of 
these  forms  of  energy  but  little  is  certainly  known;  but 
physical  investigations  have  gone  so  far  as  to  make  it 
sure  that  the  fundamental  principle  here  stated  is  true. 


117.]  ENERGY  DERIVED  FROM  THE  SUN.  121 

116.  Terrestrial  Stores  of  Energy.  From  tlie  dis- 
cussion in  the  preceding  articles  it  appears  that,  for  the 
performance  of  the  many  kinds  of  work  necessary  for 
human  life  on  the  earth,  the  best  possible  use  must  be 
made  of  the  various  forms  of  energy  which  are  ayailable, 
for  these  cannot  in  any  way  be  increased. 

The  most  important  stores  of  energy,  from  which 
mechanical  work  can  be  obtained,  are  the  following: 

1.  Energy  of  water  either  potential  or  kinetic:  this  is 
utilized  by  means  of  the  various  water-wheels,  and  made 
to  drive  mills,  etc.  This  includes  the  energy  of  tidal 
water,  which  is  also  occasionally  made  use  of. 

2.  Energy  of  wind:  employed  to  do  work  in  propelling 
ships,  and  in  turning  windmills. 

3.  Energy  of  coal,  wood,  oil,  and  other  combustibles: 
utilized  as  fuel  principally  in  the  steam-engine. 

4.  Energy  of  the  muscular  effort  of  the  various 
animals,  including  man. 

5.  Direct  energy  of  solar  heat  and  light  radiation:  it 
has  been  found  possible  to  employ  this  in  running  a 
eolar  engine,  but  thus  far  no  extensive  use  has  been 
made  of  it.  The  indirect  way  in  which  solar  heat  and 
light  have  been  and  are  still  being  utilized  is  mentioned 
in  the  next  article. 

To  the  above  may  be  added  the  energy  of  un combined 
chemical  elements,  as  sulphur  and  iron;  also,  the  internal 
heat  of  the  earth;  the  earth's  rotation  (note  the  remark 
at  the  close  of  the  next  article  on  tidal  energy);  finally, 
the  potential  energy  of  masses  of  matter  above  the  mean 
surface  of  the  earth,  which  have  been  elevated  by 
geological  changes  in  the  past. 

117.  The  Sun  as  the  Ultimate  Source  of  Terrestrial 
Energy.   All  of  the  important  forms  of  energy  just  enu- 


122  DYNAMICS.  [117. 

merated  (116)  are  derived  either  directly  or  indirectly 
from  the  sun.  The  sun  is  constantly  radiating  out  in 
all  directions  into  space  a  vast  amount  of  heat  and  light 
energy.  Of  the  whole  amount  but  a  very  minute  frac- 
tion is  received  by  the  earth,  and  only  a  very  small  part 
of  this  is  utilized,  but  this  relatively  small  amount  is 
essential  to  the  existence  of  all  kinds  of  life  on  the 
earth. 

1.  The  heat  energy  of  the  sun  causes  evaporation  from 
every  sheet  of  water;  the  water  thus  raised  in  the  form 
of  vapor  falls  again  on  the  earth's  surface,  as  rain  or 
snow,  much  of  it  at  a  level  far  above  that  of  the  sea.  It 
forms  running  streams,  or  is  collected  in  lakes  and 
ponds.  In  descending  again  to  the  sea-level  it  may  be 
made  to  do  a  great  amount  of  work. 

2.  The  heat  energy  of  the  sun  is,  also,  the  chief  cause 
in  setting  the  air  in  motion  in  the  form  of  winds,  and 
these,  as  have  been  stated,  drive  our  ships  and  turn  our 
windmills. 

3.  Still  more  important,  the  heat  and  light  energy  of 
the  sun  are  the  cause  of  all  vegetable  growth;  that  is, 
under  their  combined  action  the  chemical  change  goes  on 
by  which  the  carbon  (from  the  CO 2  in  the  atmosphere) 
is  built  up  into  the  structure  of  the  plant  or  tree.  The 
energy  thus  appropriated  is  stored  up,  but  it  may  be  ob- 
tained again,  chiefly  in  the  form  of  heat,  when  the  wood 
is  burned  as  fuel.  The  accumulated  vegetation  of  a 
former  and  far-distant  period  has  been  changed,  though 
without  any  considerable  loss  of  energy,  to  coal,  and 
this  therefore  now  represents  potentially  the  energy  of 
the  sun  received  and  utilized  by  the  earth  at  that 
time.  When,  now,  the  coal  is  burned  in  the  fire-box  of 
a  steam-engine,  this  long-stored-up  potential  energy  be- 


118.]  DISSIPATION   OF  ETiTEEGY.  123 

comes  again  kinetic,  and  from  it  we  obtain  a  large  part 
of  our  mechanical  work. 

Of  the  whole  amount  of  the  sun's  energy  which  has 
its  equivalent  in  the  resulting  vegetable  growth,  part,  as 
has  just  been  said,  is  obtained  again  in  the  combustion 
of  fuel.  Another  part  is  obtained  again  indirectly 
through  the  muscular  work  of  animals,  who  have  used 
the  vegetable  growth  in  one  form  or  another  as  food;  for 
an  animal,  as  regards  its  capacity  for  performing  physi- 
cal work,  is  to  be  regarded  as  a  machine  for  the  trans- 
formation of  energy,  which  must  be  fed  with  fuel  as 
truly  as  the  steam-engine.  A  man  forms  no  exception 
to  this  statement,  for,  in  order  that  he  may  live  and  do 
work,  he  must  also  be  fed  with  fuel;  in  his  case,  how- 
ever, the  process  is  one  step  more  complex,  since  his 
principal  food  is  the  flesh  of  animals,  who  themselves 
have  derived  their  support  from  vegetable  growth. 

The  energy  of  the  tides  must  be  made  an  exception  to 
the  preceding  remarks,  for  they  are  due  to  the  attraction 
of  the  sun  and  moon;  the  energy  of  the  tides  may,  in 
fact,  be  shown  to  be  derived  in  part  from  the  energy  of 
the  earth's  rotation,  the  rapidity  of  which  they  conse- 
quently tend  to  diminish  to  a  very  small  extent. 

118.  Dissipation  of  Energy.  If  the  illustrations  of 
the  transformation  of  energy  which  have  been  given  be 
carried  out  one  step  farther  than  is  attempted,  and  if, 
too,  the  attempt  is  made  to  apply  the  principle  of  the 
Conservation  of  Energy  to  them  rigidly,  it  will  be  seen 
that  at  each  step  in  every  transformation  there  is  an  appa- 
rent loss  of  energy  (a  real  loss  as  regards  useful  energy) 
by  its  change  into  useless  heat,  and,  moreover,  that  the 
final  form  which  it  tends  to  assume  is  always  that  of  Jieat, 

For  example,  a  pound  of  coal  produces  upon  combus- 


124  DYNAMICS.  [118. 

tion  in  the  air,  by  the  change  of  chemical  energy  into 
heat  energy,  about  7500  heat-units;  but  1390  foot-pounds 
of  work  are  the  equivalent  of  1  heat-unit;  hence  1  pound 
of  coal  should  afford  7500  X  1390  =  10,425,000  foot- 
pounds of  work.  But  the  best  steam-engines  utilize  only 
about  10  per  cent  of  this;  the  remainder  is  lost — not,  in- 
deed, as  energy,  but  so  far  as  useful  effect  goes.  Moreover, 
of  this  small  fraction  utilized,  for  example,  in  the  case  of 
the  energy  required  to  set  a  train  in  motion  and  to  carry 
it  to  the  next  station,  much  of  this  is  expended  on  the 
way  against  friction  {i.e.y  is  transformed  into  heat),  and 
when  the  brakes  are  applied  and  the  motion  of  the  great 
mass  is  arrested,  the  remainder  of  the  energy  is  also  all 
transformed  into  heat.  The  same  is  true  for  the  other 
examples  given. 

The  heat,  however,  which  is  thus  produced  as  the 
last  step  in  these  transformations  is  heat  at  a  low  tem- 
perature, which  we  are  unable  to  utilize  and  which  is 
virtually  lost.  For  as  work  can  be  obtained  from  a  body 
of  water  only  as  it  descends  from  a  higher  level  to  a 
lower,  and  if  all  were  at  the  lower  level,  no  work  would 
be  possible;  so  work  can  be  obtained  from  heat  only  as 
there  is  a  passage  from  heat  at  a  higher  to  that  at  a 
lower  temperature.  If  all  bodies  had  a  uniform  tempera- 
ture, no  work  would  be  possible.  There  is  then  a  second 
law :  that  of  the  Degradation"  of  Energy  :  according 
to  which  useful  energy  is  being  constantly  exchanged  for 
heat  of  which  no  use  can  be  made. 

EXAMPLES. 
XVIII.  Potential  and  Kinetic  Energy.     Articles  101-118. 

1.  The  weights  of  a  clock  weigh  60  lbs.  and  they  have  30  feet 
to  falb  How  much  work  do  they  represent  when  wound  up? 


118.]  ENEEGY.  125 

2.  A  mill-pond  has  a  surface  of  1  acre  and  an  average  depth 
of  6  feet;  suppose  it  200  feet  above  the  sea -level :  How  much 
potential  energy  does  it  represent  (1  cubic  foot  water  =  62.5  lbs.)? 

3.  If  the  amount  of  water  which  passes  over  a  waterfall  150 
feet  high  is  1000  cubic  feet  in  a  minute,  and  if  the  energy  derived 
from  the  fall  alone  could  be  utilized,  how  much  work-power 
would  it  represent? 

4.  If  the  same  amount  of  water,  as  in  example  3,  passes  through 
rapids  above  the  fall  at  an  average  velocity  of  16  feet  per  second, 
how  much  additional  kinetic  energy  is  here  present? 

5.  Suppose  that  the  energy  of  the  water  in  3  and  4  is  all  ex- 
pended by  the  impact  at  the  bottom  of  the  fall :  How  much  heat 
will  be  generated,  and  how  much  would  the  temperature  of  the 
water  be  elevated? 

6.  How  much  work  is  accumulated  or  stored  up  (=  kinetic 
energy)  in  a  cannon-ball  weighing  200  lbs.  and  moving  at  a  rate 
of  1200  feet  per  second?  How  much  heat  will  be  generated  if 
its  mass  motion  is  entirely  destroyed  by  the  impact  with  the 
target? 

7.  An  ounce  bullet  has  a  velocity  of  800  feet  per  second :  How 
much  work  can  it  do? 

8.  (a)  How  high  will  the  ball  in  example  7  ascend  if  moving 
vertically  upward?  (6)  How  much  farther  if  its  velocity  is 
doubled? 

9.  If  an  arrow  will  ascend  a  certain  distance  vertically,  how  far 
will  a  second  of  twice  the  weight  ascend,  the  initial  velocity  being 
the  same? 

""  10.  A  body  weighing  20  lbs.  is  projected  along  a  rough  hori- 
zontal plane  (//  =  .25)  with  an  initial  velocity  of  320  feet  per 
second :  How  far  {a)  will  it  go  before  coming  to  rest ;  how  long 
(5)  will  it  slide;  and  at  the  end  of  2  seconds  how  far  (c)  will  it 
have  gone,  and  (d)  what  will  be  its  velocity? 

11.  A  body  weighing  80  lbs.  is  projected  along  a  rough  hori- 
zontal plane  with  an  initial  velocity  of  200  feet;  the  coefficient  of 
friction  is  ^:  (a)  How  far  and  (b)  how  long  will  the  body  con- 
tinue to  move,  and  (c)  how  much  work  is  done  against  friction  in 
10  seconds? 

12.  A  body  weighing  100  lbs.  slides  down  an  inclined  plane 
whose  length,  height,  and  base  are  respectively  100,  60,  80  feet. 


126  DYNAMICS.  [118. 

and  with  the  velocity  thus  acquired  ascends  another  inclined 
plane  whose  dimensions  are  100,  80,  60  feet  respectively;  the  coeffi- 
cient of  friction  Is  .1 ;  How  far  will  it  go,  the  change  of  direction 
l)eing  supposed  to  take  place  without  loss  of  velocity? 

13.  The  body  in  example  10  is  projected  up  an  inclined  plane 
(yu  =  .25)  whose  length,  height,  and  base  have  the  ratio  of 
10:6:8     (a)  How  far  and  (b)  how  long  will  it  ascend? 

14.  An  inclined  plane  has  the  dimensions,  length  2000  feet, 
hefght  1600,  base  1200,  and  /u  =  .2:  What  velocity  of  projection 
must  a  body  weighing  64  lbs,  have  just  to  reach  the  top? 

15.  K  a  body  starting  from  rest  slides  down  the  plane  in  example 
14,  (a)  how  much  work  will  be  stored  up  in  it  when  it  reaches  the 
bottom?  Also,  (b)  how  far  will  it  slide  on  a  horizontal  surface 
(/Li  =  .2)  if  the  change  in  direction  occasions  no  loss  in  velocity? 

[In  the  following  examples  the  kinetic  energy  is  supposed  to 
be  expended  against  the  resistance  alone ;  in  fact,  a  considerable 
portion  would  result  in  the  immediate  production  of  heat.  More- 
over, the  resistance  is  supposed  to  be  uniform ;  in  fact,  this  is  not 
the  case,  and  the  result  obtained  in  each  problem  consequently  ia 
only  the  average  resistance.] 

16.  A  hammer  weighing  12  lbs.  and  moving  with  a  velocity  of 
4  feet  per  second  drives  a  nail  into  a  plank  half  an  inch:  What 
resistance  does  it  overcome? 

17.  A  weight  of  1000  lbs.,  used  as  a  pile-driver,  falls  20  feet, 
and  drives  the  pile  in  one  inch :  What  resistance  does  it  overcome? 

18.  Two  balls  weighing  100  lbs.  each  are  attached  to  the  ends 
of  a  horizontal  bar,  this  is  attached  to  a  screw  of  rapid  pitch 
(236).  They  are  made  to  rotate  rapidly  and  have  a  velocity  of 
10  feet  per  second,  when  the  end  of  the  screw  strikes  the  metal 
to  be  stamped.  Suppose  that  the  punch  comes  to  rest  after 
moving  through  ^^  inch:  What  resistance  is  overcome? 


CHAPTER  VL— STATICS. 
Introductory, 

119.  Statics  is  that  branch  of  Mechanics  which  con- 
siders the  action  of  forces  in  so  far  as  the  body  acted 
upon  is  held  by  them  in  equilibrium  (61). 

120.  Geometrical  Eepresentation  of  a  Force.  A  force 
may  be  represented  geometrically  by  a  straight  line  in  a 
manner  analogous  to  the  graphic  representation  of  velo- 
city (20).  In  each  case  (1)  the  point  of  application,  (2) 
the  direction,  and  (3)  the  magnitude  of  the  force  are 
supposed  to  be  known. 

Thus,  Fig.  46,  (1)  the  position  of  the  particle  A  acted 
upon  represents  the  point  of  application  of 
the  force;  (2)  the  direction  of  the  line  AB 
represents  the  direction  of  the  force — that 
is,  the  direction  in  which  it  tends  to  move 
the  particle  A\  (3)  the  length  of  the  line 
AB  represents  the  magnitude  of  the  force,  fig.  46. 
being  taken  proportional  to  this  magnitude  in  terms  of 
an  adopted  unit.  It  is  obvious  that  the  length  of  the 
line  has  meaning  only  when  the  unit  of  comparison  is 
known,  or  when  two  or  more  forces  are  represented  by 
lines  in  terms  of  the  same  unit;  in  the  latter  case  the 
ratio  of  the  lines  in  length  will  be  also  the  ratio  of  the 
forces  in  magnitude. 

In  statics  a  force  is  usually  exerted  as  tension  or  pres- 
sure, and  is  measured  in  pounds;  that  is,  in  accordance 


128  STATICS.  [121. 

with  the  gravitation  method  (72),  by  the  weight  it  could 
support.  The  unit  force  is  equal  to  the  force  of  attrac- 
tion exerted  by  the  earth  upon  one  pound  of  matter. 
The  possible  error  involved  in  this  method  has  already 
been  explained  (72). 

121.  Line  of  Action  of  a  Force.  The  line  of  action  of 
a  force  is  the  line  in  which  it  acts,  taken  irrespective  of 
its  direction.  Thus,  Eig.  46,  the  li7ie  of  action  of  the  force 
represented  is  the  indefinite  straight  line  AB  (or  BA) 
produced;  the  direction  of  the  force  is  from  A  to  B, 
which  is  always  indicated  by  the  order  {AB)  in  which 
the  letters  are  named,  often  also  by  an  arrowhead. 
Forces  may  therefore  have  at  the  same  time  the  same 
line  of  action  but  opposite  directions. 

122.  Transmission  of  a  Force  in  its  Line  of  Action.    It 

may  be  accepted  without  demonstration  that  if  a  force 
act  upon  a  body  at  any  point,  it  may  be  applied  without 
change  at  any  other  point  in  its  line  of  action  which  is 
rigidly  connected  with  it.  Conversely,  if  a  force  may  be 
transmitted  without  change  from  one  point  to  another 
rigidly  connected  with  it,  the  second  point  must  lie  in 
its  line  of  action. 

It  is  also  true  that  a  force  is  transmitted  by  a  string 
without  loss,  even  if  the  direction  of  the  string  is 
changed  by  its  passing  over  pegs  or  pulleys;  in  other 
words,  the  tension  of  a  string  is  the  same  at  every  point. 
In  this  statement  it  is  assumed  that  the  string  is  per- 
fectly flexible,  and  that  there  is  no  friction. 

123.  Body;  Particle.  The  definitions  of  a  preceding 
article  (2)  are  here  repeated:  A  lody  is  a  portion  of  mat- 
ter having  definite  dimensions.  A  particle  is  a  portion 
of  matter  so  small  that  any  difference  in  the  position  of 


125.]  COMPOSITION"   OF  FOECES.  129 

its  parts  may  be  left  out  of  account.  All  principles 
established  for  forces  acting  on  a  particle  will  be  true 
also  for  forces  acting  at  a  point  of  a  body,  or  for  forces 
whose  lines  of  action  produced  will  pass  through  the 
same  point  in  the  body  (122).  A  body  is  always  sup- 
posed to  be  perfectly  rigid. 

Composition  of  Forces  meeting  at  a  Point, 

124.  Composition  of  Forces.  Compokekts  and  Re- 
sultant. The  single  force  which  will  produce  the 
same  effect  as  two  or  more  forces  acting  together  is 
called  their  Resultant.  The  individual  forces  them- 
selves, which  may  be  thus  combined,  are  called  the  Com- 
ponents. 

The  process  of  iSnding  the  resultant  of  two  or  more 
forces  in  direction  and  magnitude  is  called  the  Composi- 
tion of  Forces. 

125.  General  Condition  of  Equilibrinm.  A  particle 
will  be  in  equilibrium  when  the  forces  acting  upon  it 
balance  one  another  so  that  their  resultant  is  zero.  The 
general  condition  of  equilibrium,  therefore,  for  any 
number  of  forces  acting  on  a  particle,  or  at  a  point  of  a 
body,  is 

R  =  0. 

In  order  that  this  shall  be  satisfied  various  special  con- 
ditions must  be  fulfilled,  in  the  different  cases  which 
arise,  as  mentioned  hereafter. 

It  is  obvious  that  of  several  forces  acting  on  a  particle 
and  holding  it  in  equilibrium,  each  force  will  be  equal 
and  opposite  to  the  resultant  of  all  the  others.  For  the 
remaining  forces  can  be  replaced  by  their  resultant 
without  change  of  effect,  and  in  order  that  there  shaU 


130  STATICS.  [126. 

be  equilibrium  the  first  force  must  be,  as  stated,  equal 
and  opposite  to  this  resultant.  A  body  in  equilibrium 
must  be  acted  upon  by  at  least  two  forces  ;  a  single  force 
always  causes  accelerated  motion  (60). 

126.  Composition  of  Forces  having  the  same  Line  of 
Action,  (a)  The  resultant  of  two  forces,  or  of  any  num- 
ber of  forces,  acting  in  the  same  line  and  in  the  same 
direction  is  equal  to  their  sum.  For  example,  if  P,  Q, 
S,  T,  etc.,  represent  forces  having  the  same  direction, 
as,  e.g.,  those  exerted  by  several  men  pulling  in  the 
same  line  on  a  rope,  and  R  the  resultant,  or  equivalent 
single  force,  then 

R  =  p  J^  Qj^  S-\-  T-\-  etc. 

Again,  {h)  the  resultant  of  several  forces  acting  in  the 
same  line,  but  some  in  one  and  others  in  the  opposite 
direction,  is  equal  to  the  sum  of  the  first  subtracted 
from  the  sum  of  the  second;  and  it  will  act  in  the  direc- 
tion of  the  greater  sum.  If  the  respective  directions  of 
the  forces  be  distinguished  by  their  algebraic  signs 
(+  or  — ),  then  the  resultant  will  be  equal  to  the  alge- 
hraic  sum  of  all  the  forces,  and  its  direction  will  be 
indicated  by  its  sign. 

127.  Condition  of  Equilibrium  for  Forces  having  the 
same  Line  of  Action.  The  condition  of  equilibrium  for 
two  or  more  forces  having  the  same  line  of  action  is  this: 
their  algebraic  sum  must  he  equal  to  zero. 

128.  Composition  of  two  Forces  not  having  the  same 
Line  of  Action:  Parallelogram  of  Forces.  If  two 
forces  acting  on  a  particle  he  represented  in  direction 
and  magnitude  hy  the  tivo  adjacent  sides  of  a  parallelo- 
gram, then  the  diagonal  of  this  parallelogram  passing 


128.]  COMPOSITION   OF  FOECES.  131 

through  their  point  of  intersection  will  represent  th6 
magnitude  and  direction  of  the  resultant. 

This  proposition  is  at  once  seen  to  be  closely  similar  to 
the  Parallelogram  of  Velocities  (33  and  39).  That  prin- 
ciple, as  has  been  stated  (68,  b),  is  a  deduction  from  the 
second  law  of  motion,  and  the  same  is  true  of  the  Paral- 
lelogram of  Forces.  The  part  of  that  law  upon  which 
both  propositions  are  based  may  be  stated  in  this  form: 
When  several  forces  act  simultaneously  upon  a  lody,  each 
force  produces  exactly  the  same  effect  which  it  would 
have  produced  if  it  had  acted  singly.  This  principle  is 
true  whether  the  body  was  originally  at  rest  or  in  mo- 
tion, and  it  extends  as  well  to  the  case  where  the  forces 
balance  one  another,  so  that  the  body  is  in  equilibrium. 

In  applying  the  principle  we  may  consider  the  forces 
either  {a)  dynamically,  as  producing  motion,  or  ip)  stati- 
cally, as  causing  pressure  or  tension  without  motion. 


In  the  first  case  (a),  suppose  two  forces  P  and  Q  to  act 
simultaneously  upon  the  same  particle;  each  will  have 
the  same  effect  as  if  it  acted  alone,  and  (71)  it  is  meas- 
ured by  the  velocity  it  gives  in  a  certain  time,  and  its 
direction  is  that  of  this  velocity;  therefore  if  these  velo- 
cities are  represented  (Fig.  47)  by  ^^,  AD  respectively, 
then  the  same  lines  must  be  proportional  to  the  forces 
P  and  Q)  further,  since  AC  represents  the  resultant 


132 


STATICS. 


[129. 


Telocity  in  direction  and  magnitude,  a  force  having  this 
direction  and  proportional  to  AC  must  be  the  resultant 
force  equivalent  to  the  combined  effects  of  P  and  Q. 
Hence  the  geometrical  methods  of  compounding  forces 
are  the  same  as  those  of  compounding  velocities. 

In  the  second  case  (&),  the  forces  are  generally  meas- 
ured by  the  weights  they  can  support  (72),  but  the  same 
geometrical  methods  are  also  applicable  to  them.  An 
experimental  proof  of  this  latter  case  is  given  in  the  next 
article. 

129.  Experimental  Verification  of  the  Parallelogram 
of  Forces.      Let   (Fig.  48)  A   and  B  be  two  pulleys. 


i      ^Q 


%jP 


Fig.  48. 


whose  position  may  be  changed  at  will;  over  these  are 
stretched  two  silk  threads,  knotted  at  a,  sliding  without 
friction.  At  the  extremities  of  these  threads  are  hung 
two  weights  P  and  Q,  and  from  a  is  hung  a  third 
weight,  found  by  trial  to  be  Just  sufficient  to  balance 
F  and  Q  for  the  given  position  of  A  and  B,  The  parti- 
cle a  is  now  in  equilibrium  under  the  action  of  the 
three  forces  P,  Q,  and  W,  and  it  is  obvious  that  the 
resultant  of  P  and  Q  is  equal  and  exactly  opposite  to 
Tr(125). 


130.] 


COMPOSITION  OF  FORCES. 


133 


If  now,  from  a,  ah  be  measured  off,  containing  as  many 
units  of  length  as  there  are  units  of  weight  in  Q,  and 
also  ady  containing  as  many  units  of  length  as  there  are 
units  of  weight  in  P,  these  forces  will  be  represented  in 
magnitude  by  ah  and  ad  respectively,  for  the  pulleys 
change  their  directions  only.  Complete  the  parallelogram 
ahcd',  it  will  be  found  on  trial  that  the  diagonal  ac,  which 
by  the  proposition  must  represent  the  resultant  of  P  and 
Qy  is  vertical — that  is,  directly  opposed  to  W — and  also 
that  it  contains  as  many  units  of  length  as  there  are 
units  of  weight  in  W',  hence  the  proposition  is  true  in 
this  case. 

If  the  positions  of  A  and  B  be  changed,  and  also  the 
magnitudes  of  P  and  Q,  and  a  weight  W  be  hung  at  a 
which  will  hold  the  system  in  equilibrium,  it  will  be 
found  in  every  case  that  the  proposition  is  verified  in 
the  same  manner  as  above,  and  hence  it  may  be  accepted 
as  always  true. 

130.  Calculation  of  the  Resultant.  General  Case.  Let 
the  two  forces  P  and  Q,  whose  directions  form  any 


angle  y  with  each  other,  be  represented  (Fig.  49)  by 
AB  and  AD  respectively.  Then,  if  the  parallelogram 
A  BCD  be  completed,  the  line  ^Cwill  represent  the  re- 
sultant of  these  forces.    It  is  required  to  find  an  expres- 


134 


STATICS. 


[131. 


sion  for  the  magnitude  of  this  resultant  in  terms  of 
P,  Q,  and  y. 

From  geometry  (Fig.  49), 

AC  =  AB"  +  BC  +  2AB.be,  (1) 

but  BE  =  BC  cos  CBE  =  AD  cos  BAB. 

Therefore,  substituting  the  values  of  AB,  AD,  BE  in 
(1),  we  have 

E^  =  P^  j^  Q^  ^  2PQ .  cos  r-  (2) 

The  same  formula  may  be  shown  to  hold  true  for  any 
other  case,  as  when  (Fig.  50)  the  angle  y  is  obtuse. 

-^ ?  r 


\         X 

K 

^y 

y<4^ 

^ 

P         u 
FI6.50. 

5-         M 

A^" 

P 

Fig.  51. 

Further  (Figs.  49,  50),  since  J?(7  =  AD  —  Q,  it  is  seen 
that  the  relations  between  the  two  forces  and  their 
resultant  are  equally  well  expressed  by  the  triangle  ABC. 
This  triangle  is  often  useful  for  calculation,  for  (Figs. 
49  and  51)  AB  =  P,  BC  =  Q,  AC  =  R,  BAC  =  a, 
ACB  =  DAC  =  /3,  and  ABC  =  (180°  -  DAB)  = 
(180°  —  y).  Therefore  all  the  relations  between  the 
two  forces  and  their  resultant,  in  magnitude  and  direc- 
tion, may  be  calculated  from  the  triangle  ^^C  by  the 
ordinary  methods  of  solving  an  oblique-angled  triangle. 

131.  Special  Cases,  (a)  P  =  Q.  The  general  value 
of  R  in  (2)  above  (130)  becomes,  if  P  =  Q  (Fig.  52), 


131]. 


COMPOSITIO]^^   OF  FORCES 


135 


^^  =  P»  +  P»  -f-  '-tP"  COS  y, 
=  2P'  +  2P'  cos  y, 
=  2P'  (1  +  cos  y).  ^ 


Fi3.  52. 


or 


But,  by  trigonometry,  cos  \y  —y  — ^t_ — Z^ 
1  +  cos  ;/  =  2  cos''  \y\  therefore 

E"  =  2P''  (2  cos''  \y), 
=  4:P^  cos''  iy; 
,\  E  =  2P  cos  iy. 

This  result,  as  also  those  of  (b)  and  (c)  below,  may  be 
obtained  directly  from  the  figures  without  reference  to 
the  general  case.  It  is  seen  here  that  the  resuUmit  of 
two  equal  forces  Msects  the  angle  between  them. 

(p)  P  =  Q,  y  =  60°.    The  general  value  of  E  becomes 
for  this  case 

P^  =  P»  +  P''  4-  2P''  cos  60°, 

=  3P'; 

.\E  =  PVd. 

{c)  P  =  Q,  y  =  120°  (Fig.  53).     The  general  value 
of  E  becomes  in  this  case 


E'  =  P'+P'-i-  2P'  cos  120°, 
=  2P'  -  P'; 
.-.  E  =  P. 


136 


STATICS. 


[132. 


This  result  shows  that,  if  two  forces  are  equal  and  in- 
clined at  an  angle  of  120°,  their  resultant  is  equal  to 
either  of  them.  Also,  if  three  equal  forces  acting  on  a 
particle  are  inclined  at  angles  of  120°  to  each  other,  the 
particle  will  be  in  equilibrium. 

(d)  r  =  90°  (Fig.  54).     li  y  =  90°,  or,  in   other 


words,  the  two  forces  are  at  right  angles  to  each  other, 
the  general  value  of  R  becomes 

M'  =  P'  -\-  Q\ 

This  result  is  derived  immediately  from  the  properties 
of  a  right-angled  triangle,  as  are  also  the  following  rela- 
tions: 


cos  a  = 


sm  a  = 


R' 


E' 


Q 

tan  a  =  — -, 


P  =  E  cos  a; 
Q  =  E  Bin  a; 
Q  =  P  tan  a. 


Also,  sin  a  =  cos  y5,  cos  a  =  sin  /3,  and  tan  a  =  cot  /3. 

182.  Condition  of  Equilibrium  for  Three  Forces  acting 
on  a,  Particle.     If  three  forces  acting  on  a  particle  may 


132.]  COMPOSITION   OF  FORCES.  137 

he  represented  hy  the  sides  of  a  triangle  taken  in  order, 
the  particle  will  he  in  equilihrium.  This  proposition  is 
called  the  Triangle  of  Forces. 

Let  P,  Q,  S,  represented  by  AB,  AC,  AD  respec- 
tively (Fig.  55),  be  three  forces  acting  on  a  particle  at  A, 


and  let  ahd  be  a  triangle  so  drawn  that  its  sides,  taken 
in  order,  represent  respectively  the  three  forces;  viz., 
ah  represents  P,  hd  represents  Q,  and  da  represents  8; 
then  is  the  particle  A  in  equilibrium. 

Complete  the  parallelogram  ahdc.  Since  ac  is  equal 
and  parallel  to  hd,  the  resultant  of  the  forces  ah,  ac,  will 
be  the  same  as  the  resultant  of  P  and  Q-,  hence  ad  is 
the  resultant  of  P  and  Q-,  but  da,  equal  and  opposite  to 
ad,  represents  the  third  force  S.  Therefore,  since  this 
third  force  is  equal  and  opposite  to  the  resultant  of  the 
other  two  forces,  the  particle  acted  upon  must  be  in 
equilibrium  (125). 

The  condition  "  taken  in  order"  is  essential  and  must 
be  carefully  noted. 

Cor,  The  converse  of  this  principle  is  also  true :  that 


138  STATICS.  [133. 

if  three  forces  acting  on  a  particle  keep  it  in  equilibrium, 
the  sides  of  any  triangle  which  are  respectively  parallel 
or  perpendicular  to  them  will  be  proportional  to  these 
forces. 

133.  If  three  forces  acting  on  a  particle  heep  it  in 
equililriumy  each  force  is  proportional  to  the  sine  of  the 
angle  between  the  other  two. 

Let  Py  Q,  8,  represented  by  AB,  AC,  AD  respec- 


Fio.  56. 

tively  (Fig.  56),  be  three  forces  acting  on  the  particle  A, 
and  keeping  it  in  equilibrium;  then 

P  :Q  :  8  =s>m  CAD  :  sin  BAD  :  sin  BAC 

For  take  Ahy  Ac  proportional  to  P  and  Q  respectively, 
and  complete  the  parallelogram  Aldcy  and  draw  Ad',  Ad 
will  be  proportional  to  8  and  in  the  same  straight  line 
with  it,  since  it  has  the  direction  of  the  resultant  of  P 
and  Qy  and  is  proportional  to  it.     Then 

P  '.Q:8=  Ah  '.Ac'.dA'y 

but,  by  trigonometry, 

Al)  :  Ac  {=  hd)  :  dA  --  sin  Adi  :  sin  dAb  :  sin  dbA, 

=  sin  CAD  :  sin  BAD  :  sin  BAC, 


134.]  COMPOSITION   OF  FORCES.  139 

Therefore 

P  :Q  :  S=  sin  CAD  :  sin  BAD  :  sin  BAC, 

Cor.  The  converse  of  this  proposition  is  also  true,  and 
gives  a  second  condition  of  equilibrium  for  three  forces 
acting  on  a  particle,  which  may  take  the  place  of  that  in 
Art.  132 — viz. :  If  of  three  forces  acting  on  a  particle  each 
is  proportional  to  the  sine  of  the  angle  between  the  direc- 
tions of  the  other  two,  the  particle  will  le  in  equilibrium. 
The  condition  must  be  observed,  however,  that  no  one 


of  the  forces  can  fall  between  the  directions  of  the  other 
two;  or,  in  other  words,  that  the  angular  distance 
between  no  two  of  the  forces  can  be  greater  than  180°. 
Thus  the  proportion  may  hold  good  for  the  three  forces 
in  Fig.  57,  and  also  those  in  Fig.  58;  but  only  in  the 
former  case  is  there  equilibrium. 

134.  Composition  of  more  than  Two  Forces  acting 
upon  a  Particle.  The  resultant  of  several  f urces  acting 
in  the  same  plane  upon  a  particle  may  be  found  (Fig. 
59)  by  taking  first  the  resultant  of  two  of  the  forces; 


140 


STATICS. 


[134. 


then  of  this  resultant  and  a  third  force;  again,  of  the 
resultant  of  these  three  forces  and  a  fourth  force,  and 
so  on. 

This  method  may  be  most  simply  applied  as  follows: 

Let(Fig.60)AB,AC,AD, 
AE  represent  the  four  forces 
jP,  Q,  8,  Tj  acting  on  the  par- 
ticle at  A.  From  the  point  a 
(Fig.  61)  take  ah  in  the  direc- 
tion of  P  and  proportional  to 
it;  then  in  the  same  manner 
take  he  to  represent  Q,  cd  to 
represent  8,  and  de  to  repre- 
sent T.  It  is  obvious  (com- 
pare Fig.  59)  that  ac  represents  the  resultant  of  ah,  he, 
that  is  of  P  and  Q\  also  ad  of  ac  and  cd,  that  is  of  P, 
Q,  8;  finally  ae,  the  side  which  completes  the  polygon 


Fig.  59. 


Fig.  61. 


ahcde,  is  the  resultant  of  ad  and  de,  that  is  of  the  four 
given  forces  P,  Q,  8,  T. 

The  numerical  calculation  of  the  magnitude  and 
direction  of  the  resultant  in  accordance  with  this  con- 
struction, following  the  method  already  given  (130),  in- 
volves considerable  labor.  A  more  simple  method  is 
given  in  a  subsequent  article  (140). 


136.] 


COMPOSITION   OF  FORCES. 


141 


135.  Forces  not  in  the  same  Plane.  The  method  of 
finding  the  resultant  of  any  number  of  forces  acting  on 
a  particle,  given  in  the  first  paragraph  of  the  preceding 
article,  is  also  applicable  when  the  forces  are  not  in  the 
same  plane. 

For  example,  let  AB,  AC,  AD  (Fig.  62)  represent 
the  three  forces  P,  Q,  S  respect-  ^ 

ively,  acting  on  the  particle  at  A. 
The  diagonal  AE  oi  the  parallel- 
ogram ^^^(7  will  represent  the 
resultant  of  the  forces  P  and  Q; 
also,  if  the  parallelogram  AEFD 
be  constructed,  the  diagonal  AF 
will  represent  the  resultant  of 
AE  and  AD-,  that  is,  of  P,  Q,  and 
S,  The  figure  thus  constructed 
is  sometimes  called  the  Parallelopiped  of  Forces. 

If  the  forces  are  at  right  angles  to  each  other,  then 

AE'  =  AB'  +  AC  =  P'  -{-  Q'; 

also,  AF'  =  AE'  +  AD'; 

.',  B'  =  P'  -\-  Q'  -\-  S'. 

If  a  is  the  angle  between  R  and  P,  ft  between  R  and 
Qy  y  between  R  and  8,  then 


B 


Fig. 


"IT 

62. 


cos  a  =  -^,         cos  p  =  -j^ 


cos  y 


R 


136.  Condition  of  Equilibrium  for  more  than  Three 
Forces  acting  on  a  Particle.  A^iy  number  of  forces  act- 
ing  upon  a  particle  will  liold  it  in  equilibrium,  when 
they  may  be  represented  by  the  sides  of  a  polygon  taken 
in  order.     The  forces  P,    Q,   8,  T,  and   U  (Fig.  63), 


142 


STATICS. 


[136. 


represented  by  AB,  AC,  AD,  AE,  AF  respectively, 
will  hold  the  particle  A  in  equilibrium  if  they  can 
be  represented  by  the  sides  ah,  he,  cd,  de,  ea,  taken  in 
order  of  the  polygon  ahcde.  For,  supposing  them  to 
be  so  represented,  it  has  been  shown  (134)  that  ae  is 


Fig.  63. 

the  resultant  of  the  four  forces  P,  Q,  8,  T,  and  since 
the  fifth  force,  U,  represented  by  ea,  is  equal  and 
opposite  to  ae,  it  must  hold  it  in  equilibrium ;  that  is, 
all  the  forces  must  be  in  equilibrium.  This  proposition 
is  called  the  Polygon  of  Forces.  It  obviously  applies  as 
well  to  the  case  of  forces  not  in  the  same  plane. 

EXAMPLES. 

XIX.  Parallelogram  of  Forces.     Articles  126-136. 

[The  forces  are  in  all  cases  supposed  to  act  on  a  particle,  or  at  a 

point  of  a  body  (123).] 

1.  Two  forces,  P=  7  lbs.,  Q  =  24  lbs.,  act  at  right  angles  to 
each  other;  Required  the  magnitude  and  the  direction  of  their 
resultant. 

2.  Two  forces,  P  =  13  lbs.,  Q  =  7  lbs.,  act  at  an  angle  of  138'. 
Required  the  magnitude  and  the  direction  of  the  resultant, 

3.  The  force  P=  16  lbs.  and  the  resultant  i2=24  lbs.,  and 


137.]  RESOLUTION   OF  FORCES.  143 

the  angle  between  them  is  43°:  Required  the  other  force,  Q,  and 
the  angle  between  P  and  Q  iy). 

4.  The  force  Q  =  5  lbs.  and  the  resultant  i?  =  6  lbs.;  also,  the 
angle  between  P  and  M  {a)  =  49°  30' :  What  is  the  magnitude  of 
Pand  its  direction? 

5.  Of  two  forces,  P=  16  and  Q  =  32  lbs. ;  the  angle  between  Q 
and  R  is  30° :  Required  R  and  the  angle  between  P  and  Q. 

6.  A  peg  in  a  wall  is  pulled  by  two  strings  with  forces  of  8  lbs. 
each;  they  are  equally  inclined  downward  (40°)  to  the  vertical: 
What  weight  hung  on  the  peg  would  give  an  equal  strain? 

7.  A  peg  in  a  wall  is  pulled  by  two  strings,  one  horizontal  with 
a  tension  of  21  lbs.,  and  the  other  vertical  with  a  tension  of  28 
lbs.:  What  single  force  would  exert  an  equal  pull  upon  it? 

8.  A  weight  is  supported  by  two  equal  strings  attached  to  nails 
in  the  ceiling  and  enclosing  an  angle  of  60°;  the  tension  of  each 
string  is  12  lbs.:  What  is  the  weight  supported? 

9.  Two  forces  in  the  ratio  of  3  :  4,  acting  at  right  angles  to  each 
other,  have  a  resultant  25:  What  are  the  forces? 

10.  A  boat  is  moored  in  a  stream  by  two  ropes  attached  to  the 
shore  making  a  right  angle  with  each  other;  the  tension  of  one 
{A)  is  28  lbs.,  of  the  other  (P)  is^  96  lbs.:  {a)  What  is  the  actuai 
force  of  the  current,  and  {h)  what  angles  do  the  ropes  make  with 
the  direction  of  the  current? 

11.  In  Fig.  48,  P=  12  oz.,  ^  =  15  oz.;  the  angle  hadz=m°: 
Required  W. 

12.  Of  two  forces,  P  =  2  ^,  and  they  act  at  an  angle  of  45°,  and 
P=16:  Find  Pand  Q. 

13.  Three  posts  stand  at  the  vertices  of  an  equilateral  triangle; 
a  rope  is  passed  completely  around  them,  the  tension  of  which  is 
24  lbs.:  What  is  the  pressure  on  each  post? 

\^  M  '        Resolution  of  Forces, 

137.  Resolution  of  Forces.  The  process  of  finding 
the  component  forces  whose  combined  effect  shall  be 
equivalent  to  a  given  single  force  is  called  the  Resolution 
of  Forces,  It  is  the  converse  of  the  Composition  of 
Forces. 


144 


STATICS. 


[188. 


To  resolve  a  single  force  into  two  components,  whose 
directions  are  given,  all  that  is  re- 
quired is  to  construct  a  parallelo- 
gram on  those  lines  having  the 
original  force  as  the  diagonal. 
Thus,  if  (Fig.  6^)  AC=  E  and 
the  given  directions  are  OX,  OY, 
through  G  draw  CD,  CB  parallel 
to  the  directions  given;  then^^, 

AD  will  be  the  components  required.     Similarly  for  any 

other  directions. 
For  example,  let  W  (Fig.  G5  or  QQ)  be  a  weight  hung 

by  two  strings  knotted  at  a  and  attached  at  the  points 

E  and  F,     Produce  a  W  vertically  upward,  and  let  ac 


Fia.  64. 


represent  the  weight  W,  and  through  c  draw  lines  paral- 
lel to  aE  and  aF  respectively;  then  in  the  parallelogram 
so  constructed  ab,  ad  will  be  the  components  of  the  force 
ac,  which  is  equal  and  opposite  to  W.  They  give  the 
tension  of  each  string  which  supports  the  given  weight. 

138.  Rectangular  Components.     The  case  of  the  most 

importance  in  the  resolution  of  a  single  force  is  that 

where  the  directions  of  the  two  components  are  at  right 

angles  to  each  other.     The  components  in  this  case  are 

(Fig.  67) 

AB  =  Bcosa, 

AD  =  Esm  a. 


138.] 


EESOLUTIOIT   OF  FOECES. 


145 


Here  a  is  the  angle  made  by  the  direction  of  the  first 
component  with  that  of  the  resultant.  If  the  body  is 
free  to  move  in  one  of  these  di- 
rections only,  the  component  in 
this  direction  is  called  the  effec- 
tive component,  since  this  com- 
ponent alone  influences  the  mo- 
tion of  the  body. 

For  example,  suppose  A  to  be 
a  body  either  pushed  along  (Fig. 
68)  on  a  perfectly  smooth  floor  as 
with  a  rod,  or  pulled  as  by  a  string  (Fig.  69),  the  force 
in  each  case  acting  obliquely,  as  CA  (or  A  (7).     Then  the 


Fig.  67. 


J3 -^ 


X* 


Fig.  68. 


Fig.  69. 


»|g 


effective  component,  or  that  which  alone  influences  the 
motion,  is  the  one  which  acts  in  the  direction  of  motion  ; 
that  is,  BA  or  AB  (=P  cos  /3).  The  other  component 
produces  no  effect  upon  the  motion.  It  has  already  been 
shown  that  if  the  surfaces  are  rough  and  friction  has 
to  be  considered,  the  perpendicular  component  DA 
or  AD  {=  P  sin  fi)  in  the  one  case  increases  and 
in  the  other  diminishes  the  pressure  on  the  surface, 
and  so  alters  the  resistance  of  friction  (89). 

Again  (Fig.  70),  let  «  be  a  body  resting  on  a  smooth 
inclined  plane ;  the  weight  ( W)  acts  vertically  down- 
ward {ac)f  but  the  body  is  obviously  free  to  move  only 
in  the  direction  of  the  plane.  The  weight  must  hence 
be  resolved  along  this  line  and  along  a  line  at  right 


146 


STATICS. 


[139. 


Fig.  70. 


angles  to  it ;  thus,  the  components  of  the  weight  are  ad 
and  ab,  of  which  ad  is  the 
effective  component  to  produce 
motion,  and  (the  plane  being 
perfectly  smooth)  ah  has  no 
influence  on  the  motion.  Now, 
if  HLK  =  hac  =  a,  then  ad  = 
W  sin  a,  and  ab  =  W  cos  a. 
Again,  let  a  (Fig.  71)  be  a  body  of 
weight  W  rigidly  attached  to  the 
point  0.  In  every  position  the 
weight  acts  vertically  downward, 
but  the  body  is  free  to  move  only 
in  a  direction  perpendicular  to  the 
line  of  support.  The  two  compo- 
nents of  the  weight  in  these  direc- 
tions, as  indicated  in  each  case,  are 
then  ab  =  W  cos  a,  and  ad  =  W 
sin  a,  where  a  is  the  angle  made 
with  the  vertical  direction. 

The  tension  of  the  rod,  or  the  pull 
or  push  on  the  point  of  support,  is 
always  equal  to  Wcos  a  (ab),  and  the 
effective  component  to  produce  mo- 
tion is  always  TTsin  a  {ad).  Com- 
pare the  different  positions  indi- 
cated, and  note  the  values  of  the  two 
components  in  each  of  them.  In 
positions  I  and  VII  the  moving 
component  is  zero,  and  the  tension  is  equal  to  TF ;  in  position  IV 
the  moving  component  =  TFand  the  tension  =  0. 

139.  The  explanation  of  the  fact  that  a  vessel  may  sail  in  a 
direction  almost  opposite  to  that  from  which  the  wind  is  blowing 
affords  another  illustration  of  this  principle.  Let  AB  (Fig.  72  or 
73)  represent  the  direction  of  the  wind.  The  resultant  effect  upon 
the  sail  MN  may  be  represented  by  ab.  This  force  is  resolved 
into  two  components,  one  parallel  to  the  sail  {ac  or  db)  and  pro- 


Fig.  71. 


140.] 


RESOLUTION   OF  FORCES. 


147 


ducing  no  effect,  and  the  other  perpendicular,  ad.  But  the  vessel 
is  headed  in  the  direction  IiII\  hence  to  find  the  effective  compo- 
nent of  the  wind  in  tliis  direction  the  force  ad  must  be  again  re- 
solved into  the  components  af  and  ae.  The  tendency  of  af  is  to 
drift  the  vessel  to  leeward,  and  is  nearly  balanced  by  the  resist- 
ance of  the  side  of  the  vessel  and  keel  (and  the  centre-board  in 


Fig.  72. 


Fig.  73. 


the  case  of  a  sail-boat)  against  the  water,  and  the  component  for- 
ward is  ae.  As  a  matter  of  fact  there  is  always  a  little  drifting, 
whence  the  motion  of  the  boat  is  kept  in  the  required  direction 
by  the  rudder.  The  action  of  the  rudder  is  itself  another  exam- 
ple of  the  same  principle.  It  is  seen  in  the  figures  that  with  the 
same  wind  two  vessels  may  sail  in  exactly  opposite  directions. 

An  explanation  similar  to  the  above  may  be  applied  to  the 
motion  of  a  windmill. 

140.  Resolution  of  Forces  along  Two  Axes  at  Right 
Angles  to  each  other.  The  principle  of  the  resolution  of 
a  force  along  two  axes  at  right  angles  to  each  other  may 
be  conveniently  employed  to  obtain  the  resultant  of  a 
number  of  forces  acting  at  a  common  point.  Let  the 
forces  P,  q,  S,  T  (Fig.  74)  be  represented  by  AB,  AC, 
AD,  AE,  and  let  X  and  Y  be  any  two  axes  at  right  an- 


148 


STATICS. 


[140. 


gles  to  each  other  passing  through  A,     The  components 
of  P,  Q,  S,  T  are,  geometrically, 

on  the  axis  X Ab,  —Ac,  —  Ad,  Ae,      and 

on  the  axis  Y Am,  A71,  —  Ar,  —  As, 

The  minus  sign  indicates  that  the  lines  in  question 
are  measured — that  is,  that  the  forces  act — in  the  oppo- 


Fia.  74. 

site  direction  to  the  others.  The  algebraic  sum  of  each 
set  of  these  components  will  give  the  components  of  the 
resultant  along  the  respective  axes. 

If  or',  Of",  Of"',  a^^  are  the  angles  which  each  of  the 
forces  makes  with  the  axis  X,  all  measured  in  the  same 
direction  as  indicated  in  the  figure  (74),  then  the  two 
sets  of  components  will  be  : 

F  cos  a'  +  §  cos  Of"  +  Scos  «'"  +  Tcos  a^^  =  x, 
and 

P  sin  a'  -{-  Q  sin  a''  +  Ssin  «r"'  +  ^  sin  a*^  =  y. 

The  directions  of  x  and  y  will  be  indicated  by  the  al- 
gebraic signs  belonging  to  the  sum  of  the  components  in 


14d.] 


EESOLUTION  OF  FOECES. 


149 


each  case.     If  now  the  values  of  x  and  y  be  laid  off 

from  the  point  A   (Fig.    75),   we  shall 

have,  by  completing  the  parallelogram, 

the  resultant  {R)   represented  by  AF, 

and 

and 

X^  *  Fig.  75. 


tan  p 


SO  that  the  magnitude  and  direction  of  the  resultant  are 
determined. 

141.  Condition  of  Equilibrium  for  Three  or  more 
Forces  acting  on  a  Particle.  Any  number  of  forces  act- 
ing on  a  particle  in  the  same  plane  are  in  equilihrium 
when  the  algebraic  sums  of  their  components  along  any 
two  axes  at  right  angles  to  each  other  are  equal  to  zero. 
For  then  i?  =  0,  and  this  can  only  be  true  when 

a;  =  0        and 


y 


0 


!r 


H 


but  X  is  the  algebraic  sum  of  the  components  along  one 
axis  X,  and  y  along  the  other 
axis  Y, 

This  condition  of  equilibrium 
(analytical  condition,  it  is  called) 
may  be  taken  in  place  of  that 
given  in  Art.  136. 

142.  Resolution  of  a  Force 
along  Three  Axes.  A  force  may 
be  resolved  into  components 
along  any  three  axes  not  in  the, 
same  plane.  For  example,  let  AF  (Fig.  76)  be  the  given 
force,  and  let  the  three  lines  drawn  through  A  represent 


7S 


Fig.  76. 


150  STATICS.  [142, 

the  given  axes.  Then  by  reversing  the  construction  oi 
Art.  135  the  figure  represented  in  Fig.  76  is  completed, 
in  which  AB,  AC,  AD  are  the  required  components. 

In  the  application  of  this  method  the  axes  are  ordi- 
narily taken  at  right  angles  to  each  other.  Thus,  if  a, 
P,  y  are  the  angles  between  the  given  force  (R)  and  the 
axes  X  (AB),  Y  {AC),  and  Z  {AD)  respectively,  the 
three  components  x,  y,  z  are 

X  =  R  Qo^  a,        y  =  R  Qo^  p,       z  =  R  cos  y. 

The  resultant  of  any  number  of  forces,  Bi,  R^,  Ra,  etc.,  acting 
in  different  planes  at  the  same  point,  may  be  obtained  by  carrying 
out  this  method;  for,  take  ai,  /Ji,  yi  to  represent  the  angles  made 
by  Ri  with  the  three  axes  X,  Y,  Z  respectively,  and  a^,  (5^,  y^  for 
the  angles  of  the  force  Ri  with  the  same  axes,  and  so  on ;  also,  let 
X,  y,  z  represent  the  sum  of  the  components  of  the  forces  along 
the  respective  axes ;  then 

x  =  Bx  cos  ai  +  i?2  cos  a^  r\-  B%  cos  a^  -f-  etc., 
y  =  Ri  cos  ft  I  -\-  i?2  cos  y^a  +  Ri  cos  ftz  -J-  etc., 
z  =  Bi  cos  xi  4-  ^2  cos  y^  +  Ra  cos  yt  +  etc., 
and 

Resultant  =  Vx^-]- y^-\-zK 

Also,  any  number  of  forces  acting  in  different  planes  on  a  par- 
ticle will  keep  it  in  equilibrium  if  the  sum  of  their  components 
along  any  three  axes  at  right  angles  to  each  other  is  equal  to  zero ; 
then 

x  =  0,        2/  =  0,        2  =  0. 

EXAMPLES. 
XX.  Besolution  of  Forces.    Articles  137,  138. 

1.  A  force  of  150  lbs.  is  exerted  in  a  due  north-east  direction: 
What  portion  of  it  is  felt  north?    What  portion  east? 

2.  A  weight  of  10  lbs.  (Fig.  66,  p.  144)  is  supported  by  two  strings 
of  equal  length  attached  to  nails  in  the  ceiling:  What  is  the  tension 
of  each  of  the  strings  for  the  following  angles  between  them; 
0°  (parallel).  30%  60°,  90°,  130°,  150°,  180°? 


142.]  RESOLUTION   OF  FORCES.  161 

3.  A  weight  of  20  lbs.  is  supported  by  two  strings  at  an  angle 
of  140° ;  one  (a)  goes  (Fig.  65,  p.  144)  horizontally  to  the  vertical 
wall,  and  the  other  (b)  to  the  ceiling :  What  is  the  tension  of  the 
two  strings? 

4.  If  the  angle  in  example  3  is  150°,  what  are  the  tensions  of  a 
and  5? 

5.  A  picture,  whose  weight  is  60  lbs.,  is  supported  by  a  cord 
attached  to  the  upper  corners  and  carried  over  a  nail  so  as  to 
include  an  angle  of  80°.  If  the  top  of  the  picture  is  horizontal, 
what  are  the  tensions  of  the  strings? 

6.  A  horse  drags  a  sled  by  a  rope  inclined  at  an  angle  of  15° 
with  the  ground;  the  tension  of  the  rope  is  600  lbs. :  What  is  the 
effective  component  of  the  force  exerted?  What  becomes  of  the 
other  component? 

7.  A  weight  of  18  lbs.  is  supported  by  two  strings,  one  of  which 
makes  an  angle  of  30°  with  the  vertical,  and  the  other  60° :  Find 
the  tension  of  each  string. 

XXI.  Resolution  of  Forces  along  Two  Rectangular  Axes. 
Articles  140,  141 

1.  Find  the  magnitude  and  direction  of  the  resultant  of  the 
following  forces:  P=100  lbs.,  Q  =  50,  ;S=200;  the  angle  be- 
tween P  and  Q  =  60°,  between  Q  and  8  =  120°. 

2.  Required  the  magnitude  and  direction  of  the  resultant  of 
the  following  forces:  Pz=Q  =  S=T=  100  lbs.  The  angles  are 
as  follows :  between  P  and  Q  =  30°,  between  Q  and  S  =  120°, 
between  /Sand  r=30°. 

3.  Required  the  magnitude  and  direction  of  the  resultant  of 
the  following  forces  :  P  =  Q  =  100  lbs.,  S=  T=200  lbs.  The 
angles  are:  between P and  Q  =  90°,  Q  and  >S=  135°,  S&ndT=  90. 

4.  Four  forces,  P,  Q,  S,  T,  each  equal  in  magnitude  to  100  lbs., 
have  the  following  directions  :  P=  N.  30°  E.,  Q  =  K  30°  W., 
5  =  S.  60°  W.,  r=  S.  60°  E.  What  is  their  resultant  in  direction 
and  magnitude? 

5.  Three  forces,  P,  Q,  S,  each  equal  in  magnitude  to  200  lbs., 
act  respectively  N.  45°  E.,  and  S.  45°  E.,  and  S. :  What  is  the 
direction  and  magnitude  of  their  resultant? 

6.  Three  forces,  P=  Q  =  S  =  100  lbs.,  act  respectively  E., 
N.  30°  W.,  S.  30°  W. :  What  force  will  hold  them  in  equilibrium? 


152 


STATICS. 


[ua 


7.  What  force  will  balance  the  action  of  the  four  forces  P  = 
Q  =  8=  T=100  lbs.,  acting  respectively  K  20°  E.,  N.  70°  W., 
S.  45°  W.,  S.  45°E.? 

Composition  and  Resolution  of  Parallel  Forces. 

143.  Parallel  Forces.  Forces  are  said  to  be  parallel 
when  their  lines  of  action  are  parallel.  They  are  like 
parallel  forces  if  they  act  in  the  same  direction,  and 
unlike  when  acting  in  opposite  directions. 

144.  (1)  Like  Parallel  Forces.  The  resultant  of 
two  like  parallel  forces  acting  on  a  rigid  body  is  equal  to 
their  sum,  acts  in  the  same  direction  with  them  and  at 
a  point  which  divides  the  distance  between  them  in  the 
inverse  ratio  of  the  forces. 

Let  (Fig.  77)  the  two  like  parallel  forces  P  and  Q  act 
at  the  points  A  and  B,  supposed  to  be  rigidly  connected, 


>^ 


Fig.  77. 


and  let  them  be  represented  by  ^^and  ^irrespectively. 
Also,  at  A  and  B  let  two  equal  and  opposite  forces, 
8  and  8\  be  applied;  they  will  not  change   the   con- 


144.]  PAEALLEL  FOECES.  163 

ditions,  since  they,  taken  alone,  balance  one  another. 
Find  the  resultant  AG  of  P  and  s,  also  the  resultant 
BL  of  Q  and  s',  and  produce  their  lines  of  action  till 
thev  meet  at  D.  We  may,  by  Art.  122,  suppose  them 
to  act  at  D  in  their  respective  directions.  Now  resolye 
them  (137)  into  their  components  again  in  directions  par- 
allel to  their  original  directions  :  the  components  s  (Df) 
and  s'  (DJc)  will  balance  each  other  and  may  be  disre- 
garded; and  the  other  components,  P  {Dh)  and  Q 
{Dm),  both  act  in  the  line  DG  Their  resultant  is 
therefore  equal  to  their  sum  {R  =■  P  -\-  Q),  and  may  be 
regarded  as  acting  at  the  point  G. 

Again,  since  the  triangles  DGA,  AUG  are  similar, 
and  also  the  triangles  DGB,  BML,  we  haye 


and 


Then  dividing  (1)  by  (2), 


DG      AH 
AG"  GH-~ 

P 

DG      BM 
GB"  ML" 

s' 

^y  (2), 

BG      P 

AG~  Q' 

(1) 

(2) 
(3) 


This  iBnal  equation  proves  that  the  line  ^^  is  divided 
at  G  into  segments  which  are  inversely  as  the  forces. 
From  (3),  by  inversion  and  composition,  we  obtain 

BO+AO~  P+Q'  AB       R  *' 

Ai  BO  P  BO      P  ... 

^^'    BC+AO  =  -F+q      °'     AB  =  R-        (®) 


154  STATICS.  [146. 

145.  (2)  Unlike  Parallel  Forces.  The  resultant 
of  two  unlike  'parallel  forces  is  equal  to  their  difference, 
acts  in  the  direction  of  the  greater  force,  and  at  a  point 
outside  of  it  which  divides  the  distance  between  the  two 
forces  externally  in  the  inverse  ratio  of  the  forces. 

Let  (Fig.  78)  the  two  unlike  parallel  forces  P  and  Q 
act  at  the  points  A  and  B,  rigidly  connected,  and  let. 


Fio.  78. 

them  be  represented  bj  AH  and  BM  respectively.  As 
before,  apply  two  equal  and  opposite  forces,  s  and  s',  at 
A  and  B.  Find  the  resultant  AG  ot  P  and  s,  and  the 
resultant  BL  of  Q  and  s'.  Produce  their  lines  of  action 
till  they  meet  at  I)  ;  they  will  meet  in  all  cases  unless  P 
and  Q  are  equal  (150).  Suppose  the  resultants  to  act  at 
this  point  in  their  respective  directions,  and  resolve  them 
into  components  parallel  to  the  directions  of  P  (and  Q) 
and  s.  Of  these  components  s  (Df)  and  s'  (Dh)  will 
balance  each  other,  and  Dh  {=  P)  and  Dm  {=  Q)  will 
act  at  D  in  the  same  line  and  in  opposite  directions. 


146.]  PARALLEL  FOECES.  166 

Their  resultant  will  therefore  be  equal  to  their  difference 
(or  algebraic  sum);  that  is,  in  this  case 

R  =  P-Q, 

Also,  this  resultant  may  be  regarded  as  acting'  at  C  in 
the  line  CD ;  that  is,  parallel  to  and  on  the  side  of  the 
greater  force. 

Again,  since  the  triangles  ADC  and  A  GF are  similar, 
as  also  the  triangles  BDC  and  LMB ;  then 


DC      FG      P 

AC  ~  AF~  s' 

(1) 

DC      BM      Q 
BC      ML       «'• 

(2) 

Also, 

Dividing  (1)  by  (2),  we  have 

AC  -  Q'  ^^^ 

That  is,  the  point  C  divides  the  line  AB  externally  into 
two  segments  which  are  inversely  proportional  to  the 
forces. 

Also,  we  obtain  from  (3) 

(4) 

Taking  together  equations  (3),  (4),  and  (5)  of  the  pre- 
ceding article,  and  also  the  corresponding  ones  of  this 
article,  it  is  seen  that :  Of  two  parallel  forces  and  their 
resultant y  each  force  is  proportional  to  the  distance  be- 
tween the  other  two. 


so            P 

or     ^^      ^ 

BO-AC~P-  Q' 
AC               Q 

"'     AB- K 
or     ^^       ^ 

BO-AC~P-Q' 

""     AB-R- 

156 


STATICS. 


[146. 


146.  Experimental  Verification.  The  principles  dem- 
onstrated for  like  and  unlike  parallel  forces  may  also  be 
verified  by  experiment.  (1)  Let  (Fig.  79)  xy  be  a  rigid 
rod  suspended  at  its  middle  point  C.     Also,  let  two 


X  ^ 


a 


dJt 


■  a 

ft. 


«     B 


Flo.  79. 


Op  6j? 


Fig.  80. 


weights  P  and  Q  be  taken  and  hung  on  the  bar ;  they 
are  then  two  like  parallel  forces.  It  will  be  found  that 
in  order  to  have  equilibrium  a  weight  R  equal  to  P  +  g 
must  be  hung  by  the  thread  over  the  pulley  a,  and  also 
that  P  and  Q  must  be  so  situated  that 


P 

Q 


AC 


(2)  Again,  let  (Fig.  80)  Q  be  hung  to  the  rod,  and  P 
suspended  by  the  thread  over  the  pulley ;  they  are  then 
two  unlike  parallel  forces.  In  this  case,  to  maintain 
equilibrium  R  must  be  equal  to  P  —  §.  Also,  Q  and 
R  must  be  so  situated  that 


Q       AC     .,..     P 
-^  =  -g^;thatis,-^-- 


AC 


147.  In  the  case  of  more  than  two  parallel  forces  the 
resultant  is  found  as  follows  :  First  take  the  resultant  of 
two  of  the  forces,  then  that  of  this  resultant  and  the 
third  force,  and  so  on ;  the  final  resultant  will  be  that 
of  all  the  forces  involved. 


149.]  PAEALLEL   FOECES.  157 

The  point  at  which  this  final  resultant  of  several  paral- 
lel forces  acts  is  called  the  centre  of  parallel  forces, 

148.  Three  Parallel  Forces  in  Equilibrium.  If  three 
parallel  forces  keep  a  body  in  equilibrium,  then  each 
must  be  opposite  to  the  resultant  of  the  other  two;  that 
is,  two  of  them  must  be  like,  and  the  third,  equal  to 
their  sum,  must  act  in  an  opposite  direction  at  a  point 
between  them  and  at  distances  in  inverse  ratio  to  them. 

149.  Resolution  of  Parallel  Forces.  A  single  force 
may  also  be  resolved  into  components  parallel  to  it  and 
to  each  other.  To  accomplish  this  it  is  only  necessary 
to  remember  the  rule  given  that  the  distances  from  the 
resultant  force  to  the  components  are  inversely  as  these 
forces. 

For  example,  a  weight  W  is  hung  at  a  certain  point 
Con  the  rigid  rod  AB  (Fig.  81);  it  is  required  to  find 


Fia.  81. 

the  component  pressures  P  and  Q  Sit  A  and  JB  respec- 
tively.   Divide  W  into  two  such  parts  that  P  -{-  Q  =  W, 

BC      P 
and  -j-^  =  -x-.     Or,  from  the  final  principle  in  Art.  145, 

^  ,    AB      W       ,AB       W 
take -g-^-  =  ^,  and -j-^  =  -^-. 

Again,  ABC  (Fig.  82)  is  a  table  with  a  triangular  top ;  a  weight 
Is  placed  at  a  point  0\  it  is  required  to  find  the  pressure  it  exerts 
on  each  of  the  legs  at  A,  B,  and  C.  Draw  AD,  BE,  CF,  each 
through  the  point  0;  then 

P_BF         P_EC  Q-^  (^^ 

Q~  AF'        8~AE        ^^       8~DB^         ^^ 


158  STATICS.  [150. 

But  since  the  triangles  AFC,  FBC,  as  also  AFO,  BFO,  have  the 
same  altitudes, 

BF  _  BFC     BFO  _  BOG 

AF  ~  AFC  ~  AFO  ~  AOG 


P     BOG 
'   '  Q~  AOG' 

Similarly, 

P     BOG 
S  ~  AOB' 

and 

Q      AOG 
8  "  AOB' 

Therefore 

P:  < 

)  :  8  =  BOG  :  . 

'  :  AOB.  (2) 

As  the  problem  would  ordinarily  be  stated,  the  position  of  the 
point  0  would  give  immediately  the  segments  BF,  AF,  and  BD, 


BG,   etc.,   and  therefore  equation  (1),   remembering,  also,  that 
P-\-  Q  -\-  8  =  W,  would  give  the  numerical  solution. 

If  the  point  0  is  situuted  at  the  intersection  of  the  three  lines 
drawn  from  the  vertices  to  the  middle  points  of  the  opposite  sides, 
then  obviously 

P=  Q  =  8=iW.     (See  also  166,  Gor.  1.) 

150.  Couples.  The  case  of  two  equal  and  unlike  paral- 
lel forces  is  peculiar,  since  they  have  no  resultant;  in 
other  words,  their  action  cannot  be  balanced  by  the 
action  of  any  single  force.  In  Art.  145  above,  it  F  =  Q, 


160.]  PAKALLEL  F0ECE8.  159 

then  i?  =  0;  but  when  i?  =  0,  the  valnes  oi  BC  (4)  and 
AG  (5)  become  infinite.  This  result  may  also  be  derived 
directly  from  Fig.  78,  for  as  the  difference  between  P  and 
Q  continually  diminishes  the  point  G  recedes,  and  when 
P  {AH)  =  Q  (BM)  the  lines  AG  and  BL  will  be. 
parallel,  and  therefore  G  will  be  at  an  infinite  distance. 

Two  equal  and  unlike  parallel  forces  are  called  a 
Couple.  When  acting  on  a  free  body  a  couple  tends  to 
produce  rotation,  and  the  body  can  only  be  kept  in 
equilibrium  by  the  action  of  a  second  couple  whose 
moment  of  rotation  (as  defined  below)  is  the  same  and 
in  an  opposite  direction. 

The  tendency  of  a  couple  to  produce  rotation  is 
measured  by  the  moment  of  the  couple;  this  term  will  be 
further  explained  in  the  following  article.  This  moment 
is  equal  to  the  product  of  either  force  into  the  distance 
between  them.  In  the  discussion  of  the  general  pro- 
blems which  arise  in  higher  Mechanics,  couples  play  a 
very  important  part,  but  in  this  elementary  discussion 
of  the  subject  reference  is  seldom  made  to  them,  since 
problems  involving  the  rotation  of  bodies  are  for  the 
most  part  excluded. 

EXAMPLES. 

XXII.  Parallel  Forces.    Articles  143-149. 

1.  Find  the  resultant  of  the  following  parallel  forces,  and  the 
position  of  the  point  at  which  it  acts:  (Compare  Figs.  77  and  78. ) 


(a)    P=5, 

e  =  7, 

AB  =  48. 

(&)    P=12, 

^  =  18, 

AG  =81. 

{c)    P=10, 

<2  =  -4, 

AB  =  U, 

id)   P  =  14, 

<2=-6, 

AG=^Q. 

160  STATICS.  [160. 

2  Find  the  force  Q  and  the  point  at  which  it  acts  in  the  follow- 
ing cases- 

(a)  P=S,  i?  =  8,  AB=  40. 

(6)  P  =  5,  i?  =  14,  BG  =  15. 

(c)  P  =  10,  i?  =  6,  ^i?  =  24. 

((?)  P=6,  i2  =  2,  .4(7=48. 

3.  A  rigid  rod,  supported  at  the  ends  A  and  B,  has  a  weight  of 
48  lbs  hung  6  feet  from  A  and  18  feet  from  B:  What  pressures  do 
the  supports  feel  ?  The  weight  of  the  rod  itself  is  neglected  here, 
as,  too,  in  the  following  examples. 

4.  ABC  is  a  rigid  rod ;  at  ^  a  weight  TTis  hung,  so  that  ^5=12 
and  BG  =  16;  the  pressure  at  A  is  32  lbs. .  What  is  the  pressure  at 
C,  and  what  is  W  ? 

5.  ABGD  is  a  rigid  rod;  a  weight  of  4  lbs.  is  hung  at  the  end 
A,  and  another  of  6  lbs  at  G  {AG  =  20  in.);  it  is  supported  at  B 
(BA  =  10  in.)  and  D  {DA  =  30  in.):  What  is  the  pressure  on  the 
supports  ? 

6.  A  weight  of  144  lbs  is  carried  by  means  of  a  rigid  rod  on  the 
shoulders  (at  the  same  height)  of  two  men  A  and  B;  the  distances 
from  them  are  5  and  7  feet  respectively:  What  weight  does  each 
carry  ? 

7.  A  table  has  as  its  top  an  equilateral  triangle  ABG  (Fig.  82) ;  a 
weight  ot  20  lbs.  is  placed  at  0,  so  that  the  perpendicular  distance 
from  0  on  BG  =  18  m.,  and  those  on  AG,  AB  each  equal  36  in. : 
What  is  the  pressure  on  each  of  the  three  legs  ? ' 

8.  The  top  of  a  table  is  an  isosceles  triangle  AB  =  AG=2i  feet; 
at  a  point  0,  situated  at  a  distance  of  5  inches  from  each  of  the 
equal  sides,  a  weight  of  18  lbs.  is  placed:  What  pressure  is  felt  at 
A,B,andG'}    {BAG  =  90°.) 

9.  A  rod,  whose  weight  acts  at  its  middle  point,  rests  on  two 
vertical  props  placed  at  the  ends .  Where  must  a  weight,  equal  to 
twice  that  of  the  rod,  be  placed  that  the  pressure  on  the  props 
shall  be  as  5  : 1  ? 

10.  A  rod,  whose  weight  of  18  lbs.  acts  at  its  middle  point,  is  4 
feet  long,  and  carries  a  weight  of  90  lbs.  1  foot  from  one  end : 
What  are  the  pressures  on  two  vertical  props  placed  at  the  ends  ? 


152.}  MOMENTS.  161 

Forces  tending  to  produce  Rotation — Moments, 

151.  In  all  the  cases  considered  thus  far,  the  tendency 
of  forces  to  produce  motion  of  translation  has  alone  been 
involved  (the  remarks  in  regard  to  couples  are  to  be  ex- 
cepted). We  have  now  to  do  with  forces  which  tend  to 
produce  a  motion  of  rotation. 

If  a  body  has  a  fixed  point  or  axis  and  a  force  acts  on 
it  in  any  direction  except  that  passing  through  this  point 
or  axis,  it  will  tend  to  produce  rotation  about  it.  This 
is  seen  when  a  force  acts  on  the  edge  of  a  wheel  free  to 
turn  on  an  axis. 

152.  Moment.  The  moment  of  a  force  is  the  measure 
of  the  tendency  of  a  force  to  produce  rotation  about  a 
fixed  point. 

The  moment  of  a  force  with  respect  to  any  point  may 
be  demonstrated  to  be  equal  to  the  product  of  the  force 
into  the  perpendicular  distance  from  the  point  of  rotation 
to  the  line  of  actiori  of  the  force. 

This  rotatory  effect  of  the  force  consequently  depends 
(1)  on  the  magnitude  of  the  force,  and  (2)  on  the  per- 
pendicular distance  from  the  fixed  point  upon  its  line  of 
action — or,  briefly,  upon  the  length  of  its  arm. 

For  example,  let  (Fig.  83)  a  force  P  act  at  the  point 
B  on  the  rigid  bar  ^^  to  produce  rotation  about  the 
fixed  point  A :  its  moment  is  equal  to  the  product  of  the 
force  into  its  arm;  viz., 

moment  oi  P  =  P.AB. 

This  moment  Is  increased  as  the  magnitude  of  P  is  in- 
creased, and  also  as  the  distance  AB  is  increased. 

If  the  force  acts  obliquely,  as  in  Figs.  84  and  85,  in 


162  STATICS.  [163. 

this  case  also  the  product  of  the  same  factors  gives  the 
moment  of  the  force,  but  the  arm  is  now  A  C;  that  is, 

moment  of  P  =  F.AC, 

The  same  result  would  be  obtained  if  the  effective 
component  of  F  were  multiplied  by  the  length  of  the 


Fig.  83.  Fig.  84.  Fig.  85. 

whole  line  AB.  In  the  first  case  (Fig.  84),  calhng  the 
angle  ABC  =  /3,  we  have,  as  the  moment  of  the  force, 

^ 17      7}  F.AC  =  F.AB  sin  /3;  (1) 

jpjj^^Jijj  in  the  second  case  (Fig.  86),  by  resolv- 

\  /''  ing  P,  we  have 

c 

Fig.  86.  P  sin  p.AB.  (2) 

It  is  seen  that  (1)  and  (2)  are  identical.  It  is  more  con- 
venient, and  less  likely  to  lead  to  error,  if  the  rule  given 
on  page  161  in  italics  is  uniformly  observed. 

153.  Positive  and  Negative  Moments.  As  one  force 
may  tend  to  turn  a  body  in  one  direction,  and  another 
force  in  the  opposite  direction,  it  is  necessary  to  distin- 
guish between  their  moments  in  this  particular.  This  is 
accomplished  by  calling  the  moments  jt?05^V^^'•e  (+)  where 
the  tendency  is  to  turn  the  body  in  one  direction,  and 
those  negative  (— )  which  have  the  reverse  tendency. 

154.  Geometrical  Reprssentation  of  the  Moment  of  a 
Force.     For  purposes  of  demonstration  it  is  often  con- 


155.] 


MOMENTS. 


163 


J)      JP 
Fig.  87.- 


^:b 


venient  to  consider  the  moment  of  a  force  as  repre- 
sented geometrically  by  double  the  area  of  a  triangle 
having  the  line  representing  the 
force  as  its  base  and  the  given 
point  as  its  vertex.  Thus  (Fig. 
87),  the  moment  of  the  force  P 
(AB)  about  the  point  C  is  equal 
to  AB.CD,  and  this  is  double 
the  area  of  the  triangle  ABC, 

155.  The  algelraic  sum  of  the  moments  of  two  or  more 
forces  with  respect  to  any  point  in  their  plane  is  equal 
to  the  moment  of  their  resultant. 

Let  AB,  AD  (Figs.  88,  89,  90)  represent  two  forces, 
P  and  Q;  AC,  the  diagonal  of  the  parallelogram  A  BCD 
constructed  upon  them,  will  be  their  resultant  (R).  The 
algebraic  sum  of  the  moments  of  AB  {—  AB. Eh)  and 
AD  {■=  AD. Ed),  with  respect  to  any  point  E,  is  equal 
to  the  moment  of  AC  (=  AC.Ec)  with  respect  to  the 
same  point.    There  are  three  cases  to  be  considered: 

(a)  The  point  E  falls  without  the  angles  DAB  or 


DCB  (Fig.  88),  and  hence  the  moments  of  P,  Q,  and  R 
are  all  of  the  same  kind.  The  triangle  ABE  is  equal 
to  the  sum  of  the  triangles  ADC  and  EDC,  for  they 


164  STATICS.  [155. 

have  equal  bases,  AB  and  DC,  and  the  altitude  of  the 
first  triangle — that  is,  the  perpendicular  from  U  on  AB 
— is  equal  to  the  sum  of  the  altitudes  of  the  other  tri- 
angles; that  is,  the  perpendiculars  on  the  line  DC  from 
the  Tertices  U  and  A,     Hence 

triangle  ABIJ  =  triangle  ADC  -f  triangle  BDC, 

=  triangle  ABC  —  triangle  ADB; 

,\  triangle  ABB  +  triangle  ADB  =  triangle  ABC. 

If  we  multiply  this  equation  by  2,  we  have  (by  154) 

moment  of  P  +  moment  ofQ  =  moment  of  R. 

(h)  The  point  B  falls  within  one  of  the  angles  named 
above  (Fig.  89),  and  the  moments  of  F  and  Q  are  of 


opposite  kinds.  The  triangle  ABB  is  equal  to  the 
difference  of  the  triangles  ADC,  BDC,  for  they  have 
equal  bases  AB  and  DC,  and  the  altitude  of  the  first 
triangle  (the  perpendicular  from  B  on  AB)  is  equai 
to  the  difference  of  the  altitudes  of  the  others  (the  per- 
pendiculars from  A  and  B  on  DC).     Hence 

triangle  ABB  =  triangle  ADC  —  triangle  BDC, 
=  triangle  ABD  +  triangle  ABC; 
,\  triangle  ABB  —  triangle  ABD  =  triangle  ABC, 


166.]  MOMENTS.  165 

Multiplying  this  equation  by  2,  we  obtain  '(by  154) 

moment  of  P  —  moment  ot  Q  =  moment  of  R, 

{c)  The  point  B  falls  on  the  line  of  the  resultant 
(Fig.  90).     Since  the  perpendicular  distances  from  B 


and  D  on  AC  are  equal,  the  triangles  A  ED,  AEB  have 
the  same  base  and  equal  altitudes,  and  are  therefore 
equal. 

triangle  AED  =  triangle  AEB, 

or  triangle  AED  —  triangle  AEB  =  0. 

Multiplying  by  2,  we  have 

moment  of  P  —  moment  ot  Q  =  0. 

This  is  in  accordance  with  the  proposition,  for  the 
moment  of  the  resultant  is  obviously  zero  for  this  final 
case.  The  principle  here  established  is  an  important 
one:  the  algebraic  sum  of  the  moments  of  two  forces  is 
zero  for  any  point  on  the  line  of  their  resultant. 

The  result  reached  m  this  article  may  be  readily 
extended  to  any  number  of  forces  acting  in  the  same 
plane,  whether  they  intersect  at  a  common  point  or  are 
parallel. 

166.  A  body,  free  to  turn  about  a  fixed  axis  and  acted 
upon  by  forces  171  a  plane  perpendicular  to  this  axis,  will 


166  STATICS.  [167. 

le  in  equilibrium  if  the  algebraic  sum  of  the  moments 
of  all  the  forces  about  this  axis  is  zero. 

According  to  the  condition  the  body  is  only  free  to 
rotate  in  one  plane  perpendicular  to  its  axis.  In  order 
that  it  should  be  in  equilibrium  the  tendency  to  rotate 
in  one  direction  must  be  balanced  by  the  tendency  to 
rotate  in  the  opposite  direction.  This  condition  is  satis- 
fied only  when  the  algebraic  sum  of  the  moments  of  all 
the  forces  with  respect  to  the  axis  is  zero.  By  the  con- 
cluding paragraph  in  the  preceding  article  it  is  evident 
that  the  resultant  (unless  equal  to  zero)  of  all  the  forces 
must  pass  through  this  axis,  for  only  in  this  case  can  its 
moment  be  equal  to  zero. 

This  proposition  is  a  most  important  one  and  has 
many  applications;  it  is  often  called  the  Peinciple  op 
THE  Levee. 

157.  Free  and  Constrained  Body.  A  body  which  may 
move  unrestrained  in  any  direction  is  said  to  be  free. 
On  the  other  hand,  a  body  whose  motion  is  restricted  in 
any  way  is  said  to  be  constrained. 

An  example  of  a  constrained  body  is  mentioned  in 
the  preceding  article,  and  the  condition  of  equilibrium 
for  such  a  body,  free  to  rotate  only,  is  there  given. 
Another  example  would  be  the  case  of  a  body  strung  on 
two  wires  and  free  only  to  slide  in  their  direction;  that 
is,  to  have  motion  of  translation.  The.  obvious  condi- 
tion of  equilibrium  here  is  that  the  algebraic  sum  of  the 
components  of  the  forces  taken  in  the  given  direction 
should  be  equal  to  zero. 

The  conditions  of  equilibrium  for  a  free  body,  acted 
upon  by  any  number  of  forces  in  one  plane,  require  that 
(a)  it  should  not  slide — that  is,  have  motion  of  trans' 
lation  — and  that  (b)  it  should  not  rotate. 


158.]  CONDITIONS   OF  EQUILIBKIUM.  167 

For  (a)  the  algebraic  sum  of  the  components  in  any 
two  directions  at  right  angles  to  each  other  must  be 
equal  to  zero  (141). 

Tor  (b)  the  algebraic  sum  of  the  moments  of  the  forces 
about  any  point  in  the  plane  must  also  reduce  to  zero 
(156). 

EXAMPLES. 

XXIII.  Moments.     Articles  151-156. 

1.  A  force,  P=  12  lbs,,  acts  at  right  angles  to  an  arm  6  feet 
long:  What  is  its  moment? 

2.  A  rigid  rod  AB,  8  feet  long  and  free  to  turn  about  B,  is 
acted  on  by  a  force,  P  =  64  lbs.,  whose  direction  makes  an  angle 
of  40°  with  AB:  What  is  the  moment  of  P  ? 

3.  A  force,  P=  150  lbs.,  acts  at  the  extremity  of  a  rod,  AB, 
12  feet  long,  and  at  an  angle  of  160° :  What  is  the  moment  of  P 
about  B  ? 

4.  A  bar  6  feet  long  and  pivoted  at  the  middle  has  a  weight 
of  24  lbs.  hung  at  one  extremity:  What  is  the  moment  of  the 
weight  (a)  when  the  bar  is  horizontal,  (b)  when  it  makes  an  angle 
of  40°  below,  and  (c)  of  60°  above  with  the  horizontal  position? 

[Other  examples  involving  the  moments  of  forces  are  given 
under  the  Lever.] 

Summary  of  Conditions  of  Equililrlum, 

158.  The  yarious  Cokditioits  of  Equilibrium  for 
forces  acting  on  a  body  in  one  plane,  which  hold  true 
under  the  various  circumstances,  may  be  summed  up 
here  as  follows : 

{A)  For  tioo  forces:  They  must  (1)  act  at  the  same 
point;  (2)  they  must  be  opposite;  and  (3)  they  must  be 
equal. 

{B)  For  three  forces :  They  must,  produced  if  neces- 
sary, (a)  act  at  the  same  point,  and  have  (a)  the  same 


168  STATICS.  [158. 

or  (/?)  different  lines  of  action ;   or  (h)  they  must  be 
parallel. 

(a)  a.  If  they  act  in  the  same  line,  their  algebraic 
sum  must  be  equal  to  zero  (127). 

fi.  (1)  If  they  act  at  the  same  point  and  not  in 
the  same  line,  they  may  be  represented  by  the  sides 
of  a  triangle  taken  in  order  (132);  or,  (2)  Each  will 
be  proportional  to  the  sine  of  the  angle  between  the 
directions  of  the  other  two  (133,  Cor.). 

(b)  If  parallel,  two  must  be  like  parallel  forces, 
and  the  third  must  be  equal  to  their  sum  and  act 
in  an  opposite  direction  to  them  at  a  point  distant 
from  them  in  the  inverse  ratio  of  the  forces  (148). 

(C).  For  more  tlian  three  forces : 

(a)  a.  If  they  act  in  the  same  line,  their  algebraic 
sum  must  be  equal  to  zero  (127). 

p.  If  they  act  at  the  same  point  (produced  if 
necessary),  but  not  in  the  same  line,  then:  (1)  They 
may  be  represented  by  the  sides  of  a  polygon  taken 
in  order  (136) ;  or,  (2)  The  algebraic  sum  of  their 
components  along  any  two  lines  at  right  angles  to 
each  other  mast  be  equal  to  zero  (141). 

{h)  If  they  are  parallel,  the  algebraic  sum  of  their 
moments  with  respect  to  any  point  in  the  plane 
must  be  equal  to  zero. 

(c)  If  they  act  at  different  points  or  in  different 
directions,  then:  (1)  The  algebraic  sum  of  their 
components  along  any  two  lines  at  right  angles  to 
each  other  must  be  equal  to  zero;  and  also,  (2)  The 
algebraic  sum  of  the  moments  of  the  forces  about 
any  point  in  the  plane  must  be  equal  to  zero  (157). 


CHAPTER  VIL— CENTEE  OF  GRAVITY. 


A.    CENTRE  OF  GRAVITY  OE  BODIES — PLANE  AND  SOLID. 


159.  Definition  of  the  Centre  of  Gravity.  The  attrac- 
tion of  tlie  earth  upon  all  particles  of  matter  upon  its 
surface  is  exerted  in  the  direction  of  lines  drawn  to  the 
centre.  For  the  particles  of  the  same  body,  or  of  neigh- 
boring bodies,  these  lines  may  be  regarded  as  parallel. 
For  a  given  body  the  resultant  of  all  these  parallel  forces 
will  act,  whatever  its  position,  at  a  certain  point,  called 
the  centre  of  gravity.     Hence 

TJie  centre  of  gravity  of  a  tody  is  that  point  at  which 
the  whole  weight  of  the  hocly  may  le  considered  as  concen- 
trated; or — 

It  is  a  point  at  which  the  body,  if  sujjported  there  and 
if  acted  upon  only  ly  gravity,  will  balance  in  every 
position. 

The  definition  may  be  extended  to  the  case  of  a  system 
of  bodies  if  we  suppose  them  and  their  centre  of  gravity 
to  be  rigidly  connected. 

160.  The  Centre  of  Gravity  of  Two  Bodies.     Let  P 

and  Q  (Fig.  91)  be  any  two 
bodies  of  known  weight.  It 
is  required  to  find  the  posi- 
tion of  their  centre  of  grav- 
ity. The  weights  may  be 
considered  as  two  like  paral- 
lel forces  whose  resultant  (144) 
will  be  equal  to  their  sum  and  will  act  at  a  point  which 


170 


STATICS. 


[161. 


shall  divide  the  distance  between  them  in  the  inverse 
ratio  of  the  forces.  Therefore,  if  the  straight  line  AB 
be  drawn  and  the  point  G  taken  on  it,  so  that 

AG_Q 
BG~  P' 

G  will  be  the  centre  of  gravity  of  P  and  Q.  If  this 
point  be  rigidly  connected  with  the  two  bodies,  the  sys- 
tem, supported  there,  will  balance  in  every  position. 

161.  The  Centre  of  Gravity  of  any  Number  of  Bodies. 
Let  P,  Q,  S,  and  T  (Fig.  92)  be  four  bodies  of  known 
weight   and   occupying    certain    positions  with   refer- 
ence to  each  other.     It  is 
required  to  find  their  com- 
mon centre  of  gravity.    On 
the  straight  line  AB  join- 
ing the  positions  of  P  and 
Q  take  B,  so  that 
AE_Q 
EB  ~  P' 
then,  by  160,  E  will  be  the 
centre  of  gravity  of  P  and  Q.     Again,  suppose  P  -\-  Q 
to  act  at  E,  and  on  the  line  EG  take  F,  so  that 

EF  _       8 
FG'~  P+Q' 

then  F  is  the  centre  of  gravity  of  P,  Q,  and  S.  Again, 
suppose  P  -\-  Q  -\-  S  to  net  at  F,  and  on  the  line  DFtake 
Gf  so  that 

FG  _  T 

DG~  P^Q-\-S' 

then  G  is  the  centre  of  gravity  of  the  four  bodies  P,  Q, 


Fia.  93. 


163.] 


CENTRE  OF  GEAVITY. 


171 


Sf  and  T,  and  if  it  were  rigidly  connected  with  them  the 
system  would  balance  in  every  position. 

This  method  could  obviously  be  extended,  whatever 
the  number  or  positions  of  the  given  bodies. 

162.  The  Centre  of  Gravity  of  a  Straight  Line.  T7ie 
centre  of  gravity  of  a  straight  line  is  at  its  middle  point. 
Suppose  the  line  to  be  made  up  of  a  series  of  material 
particles  of  equal  weight ;  the  centre  of  gravity  of  each 
pair  of  them  taken  at  equal  distances  from  the  centre  of 
the  line  will  be  at  this  point.  Hence  the  centre  of  grav- 
ity of  the  whole  line  will  be  at  its  centre. 

163.  The  Position  of  the  Centre  of  Gravity  of  any  Plane 
Figure  determined  by  its  Symmetry.  The  centre  of  gravi- 
ty of  any  geometrical  figure^  which  is  symmetrical  with 


reference  to  an  axis,  lies  in  this  axis.  By  a  plane  figure 
is  here  meant  any  material  geometrical  figure  whose  thick- 
ness is  uniform  and  indefinitely  small  in  reference  to  its 
other  dimensions. 

Suppose  the  figure  to  be  made  up  of  parallel  material 
lines,  all  bisected,  according  to  the  supposition,  by  the 
axis  of  symmetry.     The  centre  of  gravity  of  each  of 


172 


STATICS. 


[164. 


these  lines  (162),  and,  therefore,  of  all  of  them  taken 
together — that  is,  of  the  whole  figure — will  lie  in  this 
axis.  If  the  figure  has  two  axes  of  symmetry,  their 
point  of  intersection  will  determine  the  centre  of  gravity. 
For  example,  the  quadrilateral  A  BCD  in  Fig.  93, 
made  u|)  of  two  isosceles  triangles  placed  base  to  base,  is 
symmetrical  with  reference  to  the  axis  BD,  for  it  bi- 
sects at  right  angles  all  lines  drawn  parallel  to  AC; 
hence  the  centre  of  gravity  of  the  figure  is  in  BD.  So 
also  if,  in  Fig.  94,  AC  and  BD  are  both  axes  of  sym- 
metry, the  centre  of  gravity  must  lie  at  their  point  of 
intersection. 

164.  Centre  of  Gravity  of  Regular  Polygons.  The  posi- 
tion of  the  centre  of  gravity  of  the  regular  polygons  is 
given  immediately  by  this  principle  of  symmetry.  For 
example,  in  the  equilateral  triangle  (Fig.  95)  it  is  at  G, 
the  intersection  of  the  three  axes  of  symmetry  drawn 


y'^ 


Fig.  95, 


Fig,  96. 


Fig.  98. 


from  the  vertices  to  the  middle  points  of  the  opposite 
sides ;  in  the  square  (Fig.  96),  at  the  intersection  of 
the  lines  joining  the  middle  points  of  the  two  opposite 
sides,  or  of  the  dotted  lines  joining  the  opposite  angles ; 
in  the  regular  pentagon  (Fig.  97),  at  the  common  point 
of  intersection  of  the  five  lines,  each  drawn  from  a  ver- 
tex to  the  middle  of  the  side  opposite ;  in  the  hexagon 
(Fig.  98),  at  the  intersection  of  the  three  lines  joining 
the  middle  points  of  the  opposite  sides,  or  of  the  three 


[165.  CENTEE  OF  GEAVITY.  173 

dotted  lines  joining  the  opposite  angles;  and  so  on.  In 
the  circle,  every  diameter  is  an  axis  of  symmetry ;  the 
centre  of  gravity  is  consequently  at  the  centre. 

165.  Centre  of  Gravity  of  a  Parallelogram.  The  cen- 
tre  of  gravity  of  a  parallelogram  is  at  the  point  of  inter- 
section of  the  two  diagonals.  Let  A  BCD  (Fig.  99)  be  a 
parallelogram  whose  diagonals  intersect  at  G  ;  this  point 
is  the  centre  of  gravity.  Draw  any  line  dgb  parallel  to 
the  diagonal  DGB;  it  is  bisected  by  the  other  diagonal 
AGC.  For  from  the  similar  triangles  Adg,  ADG,  and 
Agh.AGB, 


dg       Ag 
DG~~  AG' 

and 

9^  ^9 , 
GB~  AG' 

dg        gh 
DG  ~  GB' 

or 

dg  DG 
gb  ~  GB' 

But,  by  geometry,  i)  6^  =  GB ;  hence  dg  —  gh',  therefore 
the  centre  of  gravity  of  this  line  (162)  must  be  at^y. 
Hence,  if  the  whole  figure  be  considered  as  made  up  of 


Fig.  99. 

material  lines  parallel  to  DB,  the  centre  of  gravity  of 
each — that  is,  of  the  whole  figure — must  be  in  AC.  In 
the  same  way  it  may  be  shown  to  lie  in  DB,  and  there- 
fore it  must  be  at  their  point  of  intersection  G. 

It  may  also  be  shown  that  the  same  point  is  deter- 


174  STATICS.  [166. 

mined  by  the  intersection  of  the  lines  (Fig.  100)  EF 
and  HK  joining  the  middle  points  of  the   opposite 


166.  Centre  of  Gravity  of  a  Triangle.  The  centre  of 
gravity  of  any  triangle  is  on 
the  line  drawn  from  either  ver- 
tex to  the  middle  point  of  the 
opposite  side,  and  one  third  of 
the  distance  from  this  side. 

Let  ABC  he  a  triangle  (Fig. 
101);  from  the  vertex  A  draw 
AD  to  the  middle  point  of  the 
opposite  side  BC;  also  draw 
any  line  ddc  parallel  to  BDC, 

Since  the  triangles  AM,  ABD,   and  Adc,  ADC,  are 

similar, 

hd  _  Ad  ,         dc  _  Ad 

BD~AD'        ^^^       DC~  AD' 


M        dc 

hd      BD 

or 

BD  ~  DC 

dc  ~  DC 

But,  by  construction,  BD  =  DC;  hence  hd  =  dc,  and  the 
centre  of  gravity  of  the  line  he  is  at  d,  on  the  line  AD. 
l"'herefore  it  follows  that  the  centre  of  gravity  of  all  the 
material  lines  parallel  to  BC,  of  which  the  triangle  may 
be  considered  as  made  up,  lies  in  AD,  and  hence  also 
that  of  the  whole  figure. 

Draw  from  the  vertex  B  the  line  BB  to  the  middle 
point  B  of  the  side  AC;  in  the  same  way  it  may  be 
proved  that  the  centre  of  gravity  of  the  triangle  must 
lie  in  BE.  Hence  it  must  be  at  the  intersection oi  AD 
and  BE;  that  is,  at  G, 


167.]  CENTEE   OF   GRAVITY.  175 

Connect  DE-,  since  the  triangles  AGB  and  DGE  are 
similar,  and  also  the  triangles  ^^(7  and  EDGj 

AG  _AB  AB  _BG 

GD  ~  DE'  DE  ~  ~DG' 

AG  _BG  _% 
•*•  GD~  DG~  1' 

That  is,  GD  is  one  half  oi  AG  and  one  third  of  AD. 

Cor.  1.  The  centre  of  gravity  of  three  heavy  bodies 
of  equal  weight  will  coincide  with  the  centre  of  gravity 
of  the  triangle  whose  vertices  occupy  the  position  of  the 
three  bodies.  For  (Fig.  102)  the  centre  of  gravity  of 
the  equal  weights  B  and  C  will  be  at  D,  so  that  BD  = 
DG  (160).  Also,  the  centre  of  gravity  of  B  and  G 
together  at  Z),  and  of  the  third 
weight    at  A^  will  be  at    G, 

^^  ^  AG       2        AD       S 
so  that  ^- =  3,  or  ^-^  = -. 

But  the  same  point  G  is  also 
the  centre  of  gravity  of  the 
triangle  ABC. 

Cor.  2.  The  centre  of  gravity 
of  a  polygon  can  be  found  by  dividing  it  into  triangles, 
taking  the  centre  of  gravity  of  each  by  the  above  article 
and  then  proceeding  as  in  Art.  161.  The  weights  of 
the  triangles  are  taken  as  proportional  to  their  areas. 

167.  The  Centre  of  Gravity  of  a  Solid  Figure.     The 

centre  of  gravity  of  a  solid,  which  is  symmetrical  with 
reference  to  any  plane,  must  lie  m  this  plane.  This  fol- 
lows from  the  same  consideration  as  that  in  Art.  163. 
If  there  are  two  planes  of  symmetry,  the  centre  of  gravi- 


^ 


1Y6  STATICS.  [168. 

ty  will  lie  in  their  line  of  intersection ;  and  if  three,  at 
the  point  in  which  they  all  intersect.  Therefore  the 
centre  of  gravity  of  a  sphere  is  at  its  centre ;  of  a  cylin- 
der, at  the  middle  point  of  its  axis ;  of  a  rectangular 
solid,  at  the  point  of  intersection  of  three  planes  drawn  par- 
allel to,  and  midway  between,  each  pair  of  opposite  sides. 

168.  The  Centre  of  Gravity  of  a  Triangular  Pyramid. 

TJie  centre  of  gravity  of  a  triangular  pyramid  is  on  the 
line  drawn  from  a  vertex  to  the  centre  of  gravity  of  the 
opposite  side,  and  one  fourth  of  the  distance  from  that 
side,  liet  ABCD  (Fig.  103)  be  a  triangular  pyramid. 
Take  E,  the  middle  point  of  DC,  and  draw  BE;  then  F, 
one  third  of  the  distance  on  BE  from  E,  is  (166)  the 


Fig.  103. 

centre  of  gravity  of  the  base  of  the  pyramid.  Let  hcd 
be  the  triangle  formed  by  the  intersection  of  a  plane 
drawn  through  any  point  h  parallel  to  the  base  BCD, 
Draw  AE  meeting  cd  at  e,  and  AF  intersecting  be  at 
/;  then  it  may  be  shown  that/  is  the  centre  of  gravity 


168.]  CENTRE  OF  GRAVITY.  177 

of  the  triangle  hcd.     For  from  the  similar  triangles  Ace, 

ACE,  ^udiAed,  AED, 

ce    _  Ae  ,         ed  _  Ae 

CE~AE'  ED~AE' 


ce         ed 

ce       CE 

CE  ~  ED' 

or 

ed  ~  ED' 

But  CE  =  ED',  hence  ce  =  ed,  and  e  is  the  middle  point 
of  cd.  Again,  from  the  similar  triangles  Ahf,  ABF, 
and  Afe,  AFE, 

M--M         and       A-Af. 
BF~  Ar  FE~  AF' 


•  BF      FE" 

or 

hf      BF 
fe  ~  FE' 

But  BF  =  %FE',  hence  If  —  2fe,  and/e  =  ^he,  and  /  is 
a  point  on  the  line  drawn  from  the  vertex  h  to  the  mid- 
dle point  of  the  opposite  side  one  third  of  the  distance 
from  that  side;  hence  (166) /is  the  centre  of  gravity  of 
the  triangle  hcd. 

If  now  the  whole  figure  be  thought  of  as  made  up  of 
material  triangles  all  parallel  to  BCD,  the  centre  of 
gravity  of  each  one,  and  hence  of  the  whole  pyramid, 
will  lie  in  the  line  AF.  For  the  same  reason  it  will  lie 
in  the  line  BH,  drawn  from  B  to  the  centre  of  gravity 
of  the  side  ACD-,  hence  it  will  be  at  their  point  of  in- 
tersection G, 

Since  now  the  triangles  HGF  and  BGA  are  similar, 
as  also  the  triangles  HFE  and  ABE,  we  have 
FG  _FH  FH _FE 

AG~AB'       ^"""^        AB~  BE' 
FG  _FE  _\ 
'*'  AG"  BE~  %' 


178  STATICS.  [169. 

Therefore  FG  is  one  third  oi  AG  and  one  fourth  of 
the  whole  line  AF. 

Cor.  The  centre  of  gravity  of  four  heavy  bodies  of 
equal  weight  coincides  with  the  centre  of  gravity  of  the 
triangular  pyramid  at  whose  vertices  these  bodies  are 
situated.  This  follows  in  the  same  way  as  did  Cor,  1, 
Art.  166. 

169.  To  find  the  centre  of  gravity  of  a  pyramid, 
having  any  rectilinear  polygon  as  its  base:  Divide  this 
base  into  triangles  by  lines  drawn  from  any  angular 
point  to  the  others,  and  suppose  planes  passed  through 
the  vertex  and  these  lines.  The  pyramid  is  divided 
into  a  number  of  triangular  pyramids.  The  centre  of 
gravity  of  each  of  these  will  lie  on  the  line  drawn  from 
the  common  vertex  to  that  of  its  base  and  one  fourth 
of  the  distance  from  the  base;  therefore  the  centre  of 
gravity  of  the  whole  pyramid  will  lie  in  a  plane  parallel 
to  the  base  and  one  fourth  the  distance  from  it. 

Again,  suppose  the  pyramid  made  up  of  similar  poly- 
gons parallel  to  the  base;  the  centre  of  gravity  of  each, 
and  therefore  of  the  whole  figure,  will  lie  on  a  line  drawn 
from  the  vertex  to  the  centre  of  gravity  of  the  base 
(determined  as  in  Cor.  2,  Art.  166).  The  centre  of 
gravity  of  the  pyramid  will  be  at  the  point  where  this 
line  intersects  the  plane  above  determined;  that  is,  one 
fourth  of  the  distance  from  the  base. 

170.  To  find  the  centre  of  gravity  of  a  cone:  Suppose 
the  cone  be  divided  into  an  infinite  number  of  triangular 
pyramids;  then,  by  the  reasoning  of  the  preceding  arti- 
cle, it  is  obvious  that  the  centre  of  gravity  must  lie  in  a 
plane  parallel  to  the  base  and  one  fourth  the  distance 
from  it  to  the  vertex,  and  also  in  the  line  joining  the 


171.] 


CENTRE  OF   GRAVITY. 


179 


latter  point  with  the  centre  of  gravity  of  the  base,  and 
hence  at  their  point  of  intersection. 

The  centre  of  gravity  of  the  material  surface  (taken 
in  the  same  sense  as  in  Art.  163)  of  a  right  cone  lies  on 
its  axis  and  one  third  the  distance  from  the  base  to  the 
vertex.  This  is  proved  by  showing  it  to  be  true  first 
for  a  pyramid  whose  sides  are  triangles,  and  then  pass- 
ing to  the  cone  which  is  the  limit  of  the  pyramid  when 
the  sides  are  indefinitely  increased  in  number. 

171.  Problems.  (1)  Given  the  positions  of  the  centres  of 
gravity  of  tioo  known  parts  of  a  tody,  to  find  the  centre 
of  gravity  of  the  whole.  Let  the  weights 
of  the  parts  be  w'  and  lu"  acting  at  the 
points  g'  and  g"\  then  the  centre  of 
gravity  of  the  whole  will  be  on  the  line 
^'^"  at  a  point  G  so  situated  that 

w;__g^G 
w"~  g'G' 

For  example,  suppose  the  parts  to  be 
the  isosceles  triangle  ABC  and  the 
square  BDEC,  situated  as  in  Fig.  104, 
and  \q\,  AF^=  s  (side  of  square).  The 
centre  of  gravity  of  the  triangle  is  at  g'  {g'F  =  \AF^, 

''       1 


Fig.  104. 


and  of  the  square  at^' 


W' 


(i^/'  =  i.);also,^. 


There- 


fore 


w^ 


_1  _g'*G 
~%~  g'G' 


hnt  g'g'^  =  ^s,  hence  g^G  =  ^s,  and 


(2)  Given  the  positions  of  the  centres  of  gravity  of  a 
body  and  of  a  known  part,  to  find  that  of  the  remainder. 
Let  W  be  the  weight  of  the  whole,  and  w'  of  the  part; 
then  that  of  the  remainder  =  W  —  w'  {=  w")-,  also,  let 


180  STATICS.  [171. 

the  centre  of  gravity  of  W  be  at  G,  of  w'  at  g\  and  of 
the  remainder  (w")  at  ^".     Join  g'  and  G,  and  take  on 

the  line  produced  -7^  =  —j- =  -— -;  then  is  q"  the 

^  6^/      z(?"      W  —  w'  ^ 

required  point. 

For  example,  let  the  whole  body  be  a  circle  ABC 
(Fig.  105)  whose  centre  is  G',  and  the  part  a  second 


Fig.  105. 

circle  whose  centre  is  g\  and  whose  diameter   is  the 

w'      1 
radius   {R)   of  the  larger  circle.     Then  —  =  -,  and 

and  6^^"  =  Ji2. 

EXAMPLES. 

XXIV.  Centre  of  Gramty.    Articles  159-171. 

[The  connecting  rods  mentioned  are  supposed  to  be  rigid,  and. 
except  when  otherwise  stated,  to  be  uniform  and  without 
weight.] 

1.  Where  is  the  centre  of  gravity  of  two  bodies,  A  and  B, 
weighing  4  and  5  lbs.  respectively,  rigidly  connected  by  a  weight- 
less rod  24  inches  long? 

2.  Three  weights  of  3,  6,  and  12  lbs.  are  hung,  at  ^,  B,  and  0 


171.]  CENTEE   OF  GEAVITY.  181 

respectively,  on  a  rigid  bar;  AB  =  6  inches  and  BG  =  13  inches: 
Where  will  the  bar  balance? 

3.  Four  weights  of  3,  3,  5,  and  6  lbs.,  at  A,  B,  C,  and  D,  are 
connected  rigidly  in  a  straight  line;  AB  =  10,  BG  =  8,  GB  =  18; 
Where  is  their  centre  of  gravity? 

4.  A  rod  AB,  18  inches  long  and  weighing  4  ounces,  has  a 
weight  P—  3  lbs.  hung  at  the  end  B:  Where  will  it  balance? 

5.  A  rod  AB,  34  inches  long  and  weighing  half  a  pound,  has  a 
weight  P  =  5  lbs.  at  a  point  1  inch  from  B:  Where  will  it 
balance? 

6.  What  weight  must  be  hung  at  the  end  of  a  rod  3  feet  long 
and  weighing  half  a  pound  that  it  may  balance  3  inches  from 
that  end? 

7.  A  rod  3  feet  long  and  having  a  weight  of  5  lbs.  at  one  end 
balances  at  a  point  |  of  an  inch  from  this  end:  What  is  its  weight? 

8.  A  heavy  rod,  34  inches  long  and  weighing  3  lbs.,  balances 
alone  at  a  point  10  inches  from  one  end:  What  weight  must  be 
hung  at  the  other  end  in  order  that  it  may  balance  exactly  in  the 
middle? 

9.  A  ladder  40  feet  long  and  weighing  60  lbs.  has  its  centre  of 
gravity  16  feet  from  the  larger  end :  (a)  If  supported  by  two  men, 
A  and  B,  at  the  extremities,  what  will  they  carry?  (b)  Where 
should  A  stand  to  divide  the  weight  equally  with  P? 

10.  A  uniform  rod  AB,  30  inches  long  and  weighing  3  lbs,,  has 
a  weight  of  13  oz.  at  the  end  A,  and  one  of  6  oz.  two  inches  from 
B:  Where  will  it  balance? 

11.  Weights  of  8,  4,  and  18  oz.  respectively  are  placed  at  the 
vertices  ^,  P,  C  of  a  triangle  right-angled  at  P;  AB  =18 
inches,  BG  =  9  inches :  How  far  from  G  is  their  centre  of  gravity, 
and  on  what  line? 

13.  ABG  is  an  isosceles  triangle;  AB  z=  AG=25  inches,  BG  = 
14  inches;  weights  of  4  lbs.  each  are  placed  at  A,  B,  G  respec- 
tively Where  is  their  centre  of  gravity,  measured  from  A  J 

13.  ABG  is  a  uniform  rod  bent  at  right  angles  at  P;  AB  = 
BG  =  13  inches:  Where  is  its  centre  of  gi'avity,  measured  from  P? 

14.  A  uniform  square  board,  ABGD  {AB  =  34  inches),  weighs 
8  lbs.  •  {a)  Where  will  it  balance  if  a  weight  of  1  lb.  is  placed  at 
At  (b)  if  equal  weights  of  1  lb.  each  at  A  and  P?  (c)  if  equal 
weights  of  1  lb.  each  at  A,  B,  and  G  ? 


182  STATICS.  [172. 

15.  Where  is  the  centre  of  gravity  of  the  remainder  of  a  square 
board,  ABCD  (AB  =  24  inches),  {a)  after  a  piece  is  cut  out  by 
lines  drawn  from  A  and  B  to  the  centre?  (b)  Again,  if  a  piece  is 
cut  out  by  lines  joining  the  centre  with  the  middle  points  of  two 
adjacent  sides?  (c)  Again,  if  one  corner  is  cut  off  by  a  line  join- 
ing the  middle  points  of  two  adjacent  sides? 

16.  ABC  is  an  isosceles  triangle;  AB  =  AC  =  20  inches,  BC  = 
32  inches;  the  upper  portion  is  cut  off  by  a  line  joining  the 
centres  of  these  sides:  Where  is  the  centre  of  gravity  of  the  re- 
mainder? 

17.  A  circle  having  a  diameter  of  12  inches  has  a  smaller  circle 
cut  out  of  it ;  the  diameter  of  the  latter  is  the  radius  of  the  former : 
Where  is  the  centre  of  gravity  of  the  remainder? 

18.  A  circle  has  a  diameter  of  16  inches;  a  smaller  circle  tan- 
gent to  it  and  having  a  diameter  of  12  inches  is  cut  out  of  it; 
Where  is  the  centre  of  gravity  of  the  remainder? 

19.  Find  the  centre  of  gravity  of  a  frustum  of  a  right  cone 
whose  altitude  is  8  inches,  and  the  diameters  of  the  two  bases 
6  and  12  inches  respectively. 

20.  Find  the  centre  of  gravity  of  a  figure  made  up  of  two 
isosceles  triangles  (Fig.  93,  p.  171):  BE  z=  6,  EB  =  12. 

21.  Find  the  centre  of  gravity  of  a  figure  made  up  of  a  square 
and  an  isosceles  triangle,  the  latter  having  its  base  coincident  with 
and  equal  to  a  side  of  the  square  (Fig.  104);  the  altitude  of  the 
triangle,  12  inches,  is  twice  the  side  of  the  square. 

22.  Two  uniform  cylinders  of  equal  lengths  (=  20  inches),  and 
having  diameters  of  12  and  6  inches,  are  joined  so  that  their  axes 
coincide:  Where  is  the  centre  of  gravity? 

B.    APPLICATIOJ^r   OF  THE  PRINCIPLES  OF  THE  CEI^TRE  OF 
GRAVITY — EQUILIBRIUM   AlfD   STABILITY. 

172.  Condition  of  Eq[uilibrium.  A  body  supported  at 
a  pointy  or  on  an  axis,  and  free  to  turn  about  it,  will  be 
in  equilibrium,  under  the  action  of  gravity,  if  the  verti- 
cal line  through  the  centre  of  gravity  passes  through  the 
point  or  axis  of  support.  Such  a  body  (Figs.  106,  107, 
108)  is  acted  upon  by  two  forces,  (1)  the  weight  acting 


[174. 


EQUILIBRIUM. 


183 


vertically  downward  through  the  centre  of  gravity,  and 
(2)  the  reaction  through  the  point  or  line  of  support. 
Therefore  (158,  A)  the  body  can  be  in  equilibriun  only 
as  these  forces  are  equal  and  opposite;  that  is,  the  verti- 


FlG.  106. 


Fig.  108. 


cal  line  through  the  centre  of  gravity  must  pass  through 
the  point  of  support. 

173.  Hence,  to  find  the  centre  of  gravity  of  any  body 
hy  experiment:  first  support  the  body,  as  by  a  string,  at 
one  point,  and  when  at  rest  extend  the  vertical  line 
through  the  body;  then  suspend  it  from  a  second  point, 
and  also  prolong  this  vertical  line.  The  point  of  inter- 
section of  these  two  lines  will  be  the  required  centre  of 
gravity;  for,  by  the  above  article,  the  centre  of  gravity  lies 
in  each  of  these  lines,  and  will  therefore  be  at  their  point 
of  intersection.  This  method  is  often  useful  in  the  case 
of  irregular  unsymmetrical  bodies,  to  which  the  prin- 
ciples already  given  (163,  167)  cannot  be  applied. 

174.  If  the  vertical  line  through  the  centre  of  gravity 
does  not  pass  through  the  point  or  line  of  support,  the 
body  will  tend  to  rotate  about  this  point  or  axis.  For 
(Figs.  109, 110)  the  weight  of  the  body  ABC,  represented 
by  the  vertical  line  Gd,  may  be  resolved  into  two  com- 
ponents, one,  Ga^  on  the  line  G8  drawn  to  the  axis,  and 
the  other,  Gb,  at  right  angles  to  it.    The  first  component 


184 


STATICS. 


[175. 


is  balanced  by  the  reaction  of  the  point  of  support  S, 
while  the  other  component  tends  to  make  the  body 
revolve.  The  same  result  would  be  obtained  by  taking 
the  moment  of  the  weight  about  S,  which  is  equal  to 


the  product  of  the  weight  into  the  perpendicular  dis- 
tance from  S  on  to  the  line  Gd.  This  moment  vanishes 
(155,  c)  only  when  the  condition  in  Art.  172  is  fulfilled. 

175.  Stable,  Unstable,  and  N'eutral  Equilibrium.     A 

body  is  said  to  be  in  stable  equilibrium  when,  if  slightly 
displaced,  it  tends  to  return  to  its  original  position;  it  is 


Fio.  111. 


in  unstable  equilibrium  if  it  tends  to  move  farther  from 
its  original  position;  it  is  in  neutral  Q(\m\\hrmv[i  if,  when 
moved  slightly,  it  does  not  tend  either  to  return  or  to 
move  farther. 


175.]  EQUILIBEIUM.  185 

Stable  equilibrium  is  illustrated  by  Fig.  106,  since, 
as  seen  in  Fig.  109,  a  slight  displacement  is  accompanied 
by  a  tendency  to  return  to  the  original  position.  It  is 
also  shown  in  Fig.  Ill,  where  the  weights  P,  P  bring  the 
centre  of  gravity  below  the  point  of  support.  It  is  fur- 
ther illustrated  by  a  compass-needle  as  usually  supported, 
or  by  a  hemisphere  resting  on  the  spherical  surface,  or 
a  loaded  circle  (Fig.  112).  In  the  last  two  cases  it  is  to 
be  noticed  that  the  centre  of  gravity  is  above  the  point 
of  support,  and  still  the  equilibrium  is  stable. 

Ukstable  equilibrium  is  illustrated  by  Fig.  107,  since, 
as  seen  in  Fig.  110,  a  displacement  is  accompanied  by  a 
tendency  to  rotate  farther,  the  final  position  when  the 
motion  has  been  destroyed  by  the  friction  (108)  being 
that  of  Fig.  106,  or  one  of  stable  equilibrium.  It  is  also 
illustrated  by  the  case  of  a  cone  balanced  on  its  apex,  or 


c 

Fio.  113.  Fia.  113.  Fia.  114. 

the  loaded  circle  in  the  position  shown  in  Fig.  114. 

ISTeuteal  equilibrium  is  shown  by  Fig.  108,  where  the 
centre  of  gravity  and  point  of  support  coincide.  It  is 
also  illustrated  by  the  case  of  a  cone  resting  on  its  side, 
or  of  a  sphere  in  any  position  on  a  horizontal  surface; 
also  by  a  wheel  on  an  axle  (see  also  Fig.  113). 

It  will  be  seen  that  in  stable  equilibrium  the  centre  of 
gravity  is  at  the  lowest  possible  point,  and  any  change  of 
position   of  the  body  raises  it;  it  is  on  this  account 


186 


STATICS. 


[176. 


that  the  body  tends  to  return  to  its  original  position 
when  displaced  slightly.  In  unstable  equilibrium  it 
is  at  the  highest  possible  point,  and  is  lowered  by  a 
change  of  position.  In  neutral  equilibrium  it  remains 
at  a  fixed  distance  from  the  support,  whatever  the  posi- 
tion of  the  body. 

176.  Stability  of  a  Body  resting  on  a  Base.  A  body 
resting  upon  a  base  luill  stand  or  fall  according  as  the 
vertical  line  through  the  centre  of  gravity  falls  within  or 
without  the  base  of  support.  By  base  of  support  is  meant 
the  salient  polygon  formed  by  lines  joining  the  extreme 
points  of  support.  For  example,  for  a  table  with  three 
legs  it  is  a  triangle  formed  bylines  joining  their  extremi- 
ties. 

This  principle  is  illustrated  in  Figs.  115,  116,  117, 


Fio.  115. 


Fig.  116. 


Fig.  118. 


118.  In  each  case,  let  a  vertical  line  through  the  centre 
of  gravity  {G)  represent  the  weight  of  the  body;  then, 
taking  the  moment  (152)  of  the  weight  about  the  point 
C,  which  would  be  the  axis  in  case  of  an  overturn  in 
that  direction,  the  product  of  Wx  EC  measures  the  ten- 
dency of  the  body  to  retain  its  position  (Figs.  115,  116), 
and  the  product  Wx  CE,  Fig.  118,  measures  the  tendency 
of  the  body  to  overturn.  In  Fig.  117  the  line  of  the 
weight  passes  through  the  axis  of   rotation;  hence  its 


177.] 


STABILITY. 


187 


moment  is  zero,  and  the  body  is  on  the  point  of  over- 
turning. 

In  general,  when  the  vertical  line  of  *  the  weight  falls 
within  the  base,  the  product  of  the  weight  into  the  per- 
pendicular distance  to  the  nearest  side  is  called  the  mo- 
ment of  stability.  When  the  same  line  falls  without,  this 
product  of  the  weight  into  the  perpendicular  distance 
to  the  nearest  side  is  called  the  moment  of  instahility. 

177.  Conditions  upon  which  the  Stability  of  a  Body 
depends.  If,  as  in  Figs.  119,  120,  a  force  P  acts,  as  indi- 
cated, at  the  point  A,  the  body  will  be  on  the  point  of 
overturning  when  the  moments  of  P  and  W  about  C  are 
equal  and  opposite  (156).  If,  in  general,  the  arm  of  P 
is  R  (here  BC),  and  of  PTis  r  {EC),  then 

PXE=  Wx  r. 

Also,  if  a  force  P  acts  to  support  a  body  tending  to  over- 
turn (as,  for  example,  a  prop),  then  the  same  equation 


^•^ 

2?^/ ■ 

W 

L 

—3^ 

> 

\s 

Fig.  119. 


Fio.120. 


Fio.  121 


will  hold  true  when  the  body  is  supported,  as  shown  in 
Fig.  122,  and  the  value  of  P  given  by  the  equation  is 
the  pressure  on  the  prop. 

In  the  first  case  it  is  obvious  that  the  greater  the  over- 
turning force  required,  the  greater  the  stability  of  the 
body.  But  from  the  above  equation,  P  must  increase  as 
IT  increases;  that  is — 


188 


STATICS. 


[177. 


(1)  The  stability  is  greater  as  tlie  weight  increases 
(other  conditions  being  equal);  e.g.,  a  stone  tower  is  less 
easily  blown  over  than  a  wooden  one  of  the  same  dimen- 
sions. 

Also,  P  increases  as  r  increases;  that  is — 

(2)  The  greater  the  base  of  support  the  greater  the 
stability,  other  conditions  remaining  the  same.  Also,  if 
the  base  is  a  regular  polygon  of  given  area,  supposing 
the  vertical  line  to  fall  through  the  centre,  the  stability 
is  least  if  it  is  a  triangle,  it  increases  with  the  number 
of  sides,  and  is  greatest  for  the  circle. 

Still  again,  if  W  and  r  are  constant,  P  is  increased  as 
E  diminishes,  and  conversely;  in  other  words — 

(3)  The  greater  the  arm  of  the  power  the  more  readily 
is  the  body  put  on  the  point  of  overturning.  For  exam- 
ple, of  two   towers  of  different  heights  but  otherwise 


Fio.  123. 


alike,  the  higher  would  be  the  more  easily  blown  over, 
since  the  resultant  force  of  the  wind  would  act  at  a 
greater  distance  from  the  base. 

Finally — (4)  The  stability  is  increased  as  the  position 
of  the  centre  of  gravity  is  lowered.  This  relation  does 
not  follow  from  the  equation,  since  that  applies  only  to 
the  case  of  equilibrium,  where,  as  it  was  expressed,  the 
body  is  on  the  point  of  overturning. 


177.] 


STABILITY. 


189 


In  order  that  the  body  should  actually  be  overturned 
P  must  continue  to  act  (Figs.  123,  124,  125),  diminish- 
ing continually  as  r  diminishes,  and  becoming  zero  when 
the  body  is  in  the  second  position  indicated  in  each 


Fia.  125. 

figure.  Hence  work  is  done  in  accomplishing  this  re- 
sult, and  this  is  estimated  most  simply  by  the  product 
of  the  weight  into  the  distance  which  the  centre  of 
gravity  is  raised  (97).  The  work  done  increases  as  the 
position  of  the  centre  of  gravity  is  lowered.  For  ex- 
ample, the  initial  values  of  P  are  the  same  in  Figs.  123, 
124,  and  125,  as  also  the  final  values  (=  0),  but  the  arc 
Aa,  through  which  P  acts,  and  the  height  Fg,  through 
which  the  weight  is  raised,  are  least  for  the  highest 
position  of  the  centre  of  gravity  (Fig.  123),  and  greatest 
for  its  lowest  position  (Fig.  125).  Thus,  a  stage-coach 
with  a  heavy  load  of  trunks  on  top  has  its  centre  of 
gravity  high,  and  is  easily  overturned  by  a  slight  irregu- 
larity in  the  road. 

EXAMPLES. 
XXV.  StaMlity.    Articles  172-177. 

[The  centre  of  gravity  in  each  case  is  assumed  to  be  at  the  geo- 
metrical centre,] 
1.  If  the  weight  of  the  structure,  of  which  ABGD  (Fig.  115, 
p.  186)  is  a  section,  is  150  lbs.,  also  ABz=  6  feet,  AD  =  10  feet: 


190  STATICS.  [177. 

What  force  P  will  put  it  on  the  point  of  overturning  {a)  if  acting 
at  ^?  {h)  if  acting  at  the  centre  of  ADt  (c)  If  it  rests  on  the  side 
AD  and  the  force  acts  at  B  and  the  middle  point  of  AB  respec- 
tively, what  answers  are  obtained? 

2.  In  Fig.  132,  p.  187,  AD  =  AB  =  10  feet,  the  angle  ABC  =  90° 
and  BGE  =  15°,  the  weight  is  120  lbs. :  What  is  the  pressure  on 
a  prop  placed  {a)  so  as  to  act  at  B  at  right  angles  to  BG  ?  (b)  so  as 
to  act  at  the  same  point  but  standing  vertically? 

3.  A  rectangular  frame  ABGD  (Fig.  117,  p.  186)  is  racked  out 
of  shape.  If  AB  =  12  feet,  AD  =  18  feet,  what  is  the  angle  ADG 
when  it  is  about  to  fall? 

4.  A  uniform  stone  tower,  8  feet  in  diameter,  inclines  1  foot 
for  every  10  feet  of  vertical  height :  What  is  the  height  of  the  top 
when  it  is  about  to  fall? 

5.  {a)  A  rough  plane  is  inclined  so  that  a  cube  resting  on  it  is 
about  to  turn  over,  it  not  being  able  to  slide:  What  is  the  angle? 
(6)  What  is  the  angle  for  an  isosceles  triangle  {AB=^  AG=\0, 
BG  =  16)  if  it  rests  on  the  side  AB  ? 

6.  A  table  6  feet  square  stands  upon  four  legs,  each  of  which  is 
12  inches  in  from  the  adjacent  edges;  its  height  is  3  feet  and  its 
weight  24  lbs. :  What  is  the  least ,  force  required  to  put  it  on  the 
point  of  overturning  if  applied  at  the  edge  (a)  as  a  horizontal 
push?  (b)  as  a  pressure  directly  down? 

7.  A  table,  having  a  circular  top  of  2  feet  radius,  is  supported 
on  three  legs  placed  at  the  edge  and  at  equal  distances  from  one 
another;  the  height  is  30  inches  and  the  weight  20  lbs. ;  What  is 
the  least  force  that  will  put  it  on  the  point  of  overturning  if  applied 
at  the  top  (a)  as  a  horizontal  push?  (b)  as  a  pressure  down?  (c)  act 
ing  vertically  upward? 

8.  What  work  would  be  done  in  overturning  a  cylindrical 
column  of  stone  weighing  40,000  lbs.,  10  feet  high  and  4  feet 
diameter,  supposing  that  the  centre  of  gravity  is  on  the  axis  (a;  <i,t 
the  middle?  {b)  1  foot  from  bottom?  (c)  1  foot  from  top? 


CHAPTEK  VIIL— MACHINES. 

178.  The  MACHINES  are  mechanical  contrivances,  by 
the  use  of  which  a  force  applied  at  one  point  is  made  to 
act  at  another  with  a  change  in  either  its  direction  or 
intensity,  or  in  both.  By  means  of  them,  for  example, 
the  power  may  raise  a  weight  much  larger  than  itself, 
or,  on  the  other  hand,  it  may  give  to  the  weight  a 
velocity  much  greater  than  its  own.  In  all  cases,  how- 
ever, the  machine  is  only  an  instrument  by  which  me- 
chanical energy  is  transformed;  it  never  creates  energy. 

179.  The  principle  of  the  preceding  article  was  laid 
down  in  Art.  98,  where  it  was  stated  that,  as  follows 
from  the  law  of  the  Conservation  of  Energy,  in  every 
case:  "The  work  done  by  the  power  is  equal  to  the 
work  expended  upon  the  weight." 

The  work  done  by  the  force  (P)  acting  is  equal  to  the 
product  of  it  (or  its  effective  component,  P  cos  j3*)  into 
the  distance  {s)  through  which  it  acts;  that  is, 

P.s,        or        P  cos  /3.S.  (1) 

The  work  done  in  raising  the  weight  is  equal  to  the  pro- 

*  When  the  force  acts  obliquely  to  the  motion  of  the  body,  it  is 
immaterial,  in  the  estimation  of  the  work  done,  whether  the  pro- 
duct of  the  effective  component  of  the  force  into  the  whole  distance 
is  taken  (=  P  cos  ^.s),  or  the  product  of  the  whole  force  into  the 
effective  distance;  that  is,  the  resolved  part  of  the  motion  in  the  di- 
rection of  its  own  action  (=  P.s  cos  /3). 


192  STATICS.  [180. 

duct  of  it  (W)  into  the  vertical  distance  (h)  through 
which  it  is  raised;  that  is, 

W.h.  (2) 

If  the  work  is  done  not  against  gravity  in  raising  a 
weight,  but  against  some  other  force  producing  a  resis- 
tance, the  work  done  is  estimated  by  the  resistance  over- 
come (R)  into  the  effective  distance  (d);  that  is, 

B.d.  (3) 

It  is,  in  general,  found  convenient  to  use  the  term  weight 
as  including  the  resistance,  though  the  true  distinction 
must  not  be  forgotten. 

In  every  machine,  according  to  the  principle  of  work 
just  stated, 

F.s  =  WJi,    and       P.s  =  R,d,  (4) 

The  relation  (5)  may  be  expressed  in  this  way,  that: 
Tlie  Poioer  is  to  the  Weight  (or  Resistance)  as  the  dis- 
tance through  which  the  Weight  is  raised  (or  the  Resist- 
ance is  overcome)  is  to  the  distance  through  which  the 
Power  acts. 

180.  Machines  are  then  employed:  (1)  Where  a  small 
power  is  desired  to  raise  a  large  weight  or  overcome  a 
great  resistance.  In  this  case  there  is  said  to  be  a 
mechanical  advantage;  but  as  seen  from  equation  (4), 
if  W  is  greater  than  P,  s,  the  distance  through  which 
P  acts,  must  be  as  many  times  greater  than  A,  the  dis- 
tance through  which  W  rises.  This  is  sometimes  ex- 
pressed in  this  form:  What  is  gained  in  power  is  lost  in 
velocity. 


182.]  MACHINES  IN  GENEEAL.  193 

Also — (2)  Where  an  increased  velocity  is  required;  in 
this  case  h  (or  d)  will  be  greater  than  s,  but  P  must  be 
as  many  times  greater  than  W.  There  is  then  said  to  be 
a  mechanical  disadvantage,  but,  similar  to  the  principle 
above,  what  is  lost  in  potver  is  gained  in  velocity. 

The  cases  where  the  attention  is  directed,  in  the  use 
of  the  machine,  solely  to  the  diminished  or  increased 
velocity  of  the  motion  it  transmits,  the  relation  of  P  to 
W  being  overlooked,  are  obviously  included  in  (1)  or  (2). 

(3)  Machines  are  also  occasionally  employed  where 
only  change  in  direction  is  required,  and  here  P  —  W  {or 
R),  and  hence  s  =  h  (or  d), 

181.  The  relation  of  P  to  IF",  established  in  equation 
(4),  is  that  wliich  is  required  in  order  that  the  power 
acting  uniforynly  should  raise  the  weight  uniformly.  If 
the  value  of  the  power  were  greater  than  that  thus  re- 
quired, accelerated  motion  would  ensue;  and  if  less,  there 
would  be  retarded  motion  and  the  system  would  ulti- 
mately come  to  rest. 

The  same  relation  of  P  and  W  will  hold  good  if  the 
system  is  at  rest  and  the  power  simply  supports  the 
weight.     The  principles  of  statics  make  it  possible  to 

P 
deduce  independently  this  ratio  of  -^  on  the  supposi- 
tion that  the  weight  is  at  rest.  In  the  pages  which 
follow,  the  relation  will  be  deduced  by  both  methods: 
first  in  accordance  with  statics,  and  second  on  the  prin- 
ciple of  work. 

182.  Virtual  Velocities.  In  the  second  case  given  above, 
the  principle  of  work  may  be  stated  in  this  form: 
If  any  machine,  in  equilibrium  under  the  action  of 
several  forces,  suffers  a  slight  displacement  consistent 


194  STATICS.  [183. 

with  tlie  relations  of  the  parts,  then  the  algebraic  sum 
of  the  work  done  by  the  forces  will  be  zero,  and  con- 
versely. For  such  a  case  as  this  the  velocities  are 
imaginary,  and  are  called  virtual.  This  is  sometimes 
spoken  of  as  the  principle  of  viktual  velocities.  This 
principle  is  essentially  that  involved  in  equation  (4)  or 
(5)  of  Art.  179. 

183,  The  Machines  with  Friction.  In  the  statements 
which  have  been  made  in  regard  to  the  relation  of  the 
power  and  weight  in  the  case  of  a  machine,  it  has  been 
assumed  that  the  work  done  by  the  power  was  all  ex- 
pended in  raising  the  weight.  In  practice,  however, 
there  are  various  hurtful  resistances  to  be  overcome, 
chief  among  which  is  friction.  Hence  the  work  done 
by  the  power  must  always  be  greater  than  that  which 
the  equation  (4)  requires.  For  example,  if  i^  represents 
the  force  of  friction,  and  I  the  distance  through  which 
it  is  overcome,  then  the  work  done  against  friction,  as 
shown  in  Art.  100,  is  F.l^  and  the  equation  must  then 
be  written 

P.s  =  W.n^F.l. 

The  law  of  the  Conservation  of  Energy  still  holds  good; 
but  as  the  term  F.l  increases,  the  amount  of  work  ex- 
pended in  producing  no  useful  effect,  but  merely  use- 
less heat  (110),  is  increased.  Hence,  although  there  is 
theoretically  no  limit  to  the  mechanical  advantage  that 
may  be  attained  (though  always  with  a  proportional  loss 
of  velocity)  by  an  appropriately  constructed  machine  or 
combination  of  machines,  there  is  practically  a  limit; 
for,  as  the  complexity  increases,  more  and  more  of  the 
power  is  expended  without  useful  effect. 

To  the  resistance  of  friction  must  be  added  other  re- 


185.]  LEVEE.  195 

sistances  which  are  additional  drains  upon  the  energy 
communicated  to  the  machine,  and  which  leave  less  to 
be  expended  in  raising  the  weight.  Among  these  are: 
adhesion  of  parts  in  contact,  the  stiffness  of  cords,  resist- 
ance of  the  air,  want  of  rigidity  in  the  parts  of  the  ma- 
chine. In  the  discussion  in  the  following  pages  all  these 
resistances  are  left  out  of  account.  The  weights  of  the 
parts  of  the  machines  are  also  to  be  neglected  unless 
otherwise  stated. 

The  modulus  of  a  machine  is  the  ratio  of  the  amount 
of  work  practically  done  by  it  to  that  which  theory  re- 
quires. 

184.  Simple  Machines.  The  Simple  Machines,  or 
Mechanical  Powers  as  they  are  sometimes  called,  are 
as  follows:  (1)  The  Lever,  (2)  Wheel  and  Axle,  (3) 
Toothed  Wheels,  (4)  Pulley,  (5)  Inclined  Plane,  (6) 
Wedge,  (7)  Screw. 

Of  these  machines  the  Wheel  and  Axle  and  Toothed 
Wheels  are  in  fact  modifications  of  the  Lever.  All  of 
them  involve  the  essential  idea  of  a  tendency  to  rotation 
about  an  axis,  and  hence  to  deduce  the  conditions  of 
equilibrium  for  them  the  principle  of  the  equality  of  mo- 
ments is  employed  (156).  The  Pulley  is  based  upon  the 
principle  of  reduplication,  depending  on  the  fact  of  equal 
transmission  of  force  by  a  string ;  or,  in  other  words, 
that  the  tension  of  a  rope  at  every  point  is  the  same 
(122).  The  Wedge  and  Screw  are  essentially  modifica- 
tions of  the  Inclined  Plane. 

I.  Leyer. 
A.   General  Principle  of  the  Lever, 

185.  The  Lever  in  its  simplest  form  is  a  rigid  bar 
capable  of  being  turned  about  a  fixed  axis  called  the  ful- 


196  STATICS.  [186. 

crum,  and  supposed  to  be  acted  upon  by  forces  in  a  plane 
at  right  angles  to  this  axis.  The  bar  may  have  any  shape, 
straight,  bent,  or  curved,  and  the  directions  in  which  the 
power  and  weight  act  may  make  any  angles  with  it. 

186.  For  all  forms  of  the  lever  the  condition  of  equi- 
librium is  that  stated  in  Art.  156  for  a  constrained  body 
only  free  to  move  about  an  axis  in  a  plane  at  right  angles 
to  it.  The  power  tends  to  produce  rotation  about  the 
fulcrum  in  one  direction,  and  the  weight  in  the  other. 
Hence — 

If  the  Poioer  and  the  Weight  are  in  equilibrium,  the 
moment  of  the  Foiver  must  be  equal  and  opposite  to  the 
moment  of  the  Weight. 

This  may  also  be  stated  as  follows  : 

The  Foiuer  is  to  the  Weight  as  the  perpe^idicular  dis- 
tance from  the  fulcrum  to  the  direction  of  the  Weight 
is  to  the  perpendicular  dista^ice  from  the  fulcrum  to  the 
direction  of  the  Power, 

For  example,  m  Figs.  126,  127,  128,  where  the  bar  is 


c 


J'  s 


^ 


Fig.  126.  Fio.  127.  Fig.  128. 


straight  and  the  power  and  weight  act  at  right  angles 
to  it, 

P.AF=    W.BF, 

P  _   BF 
^^  W  ~   AF' 

The  rule  holds  good  equally  well  when  the  bar  is  not 
straight  and  the  directions  are  oblique.     The  positions 


186.] 


LEVEE. 


197 


of  the  perpendicnlar  distances  from  the  fulcrum — that 
is,  the  arms  of  P  and  Tl^'^— are  to  be  carefully  noted  in 
the  following  figures,  129-134.  For  all  of  them  the 
same  equation  holds  good. 

The  pressure  on  the  fulcrum  (the  weight  of  the  lever 
being  neglected)  in  Figs.  126  and  129  is  P  +  W,  in  Fig. 
127  it  is  W-  P,  in  Fig.  128  it  is,  P  —  W,  and  in  the 
other  figures  it  may  be  calculated  by  the  parallelogram 
of  forces.  In  the  latter  cases  it  is  to  be  noted  that, 
since  the  power,  weight,  and  resistance  of  the  fulcrum 
are  in  equilibrium,  their  lines  of  action  produced  must 
pass  through  the  same  point  (158,  P). 


Fio.  129. 


Fia.  131. 


In  Fig.  132  the  lever  is  curved  like  an  iron  pump- 
handle,  the  arms  of  the  weight  and  power  are  the  per- 


rw 


Fia.  132. 


pendiculars  BF  and  AF  respectively,  and  the  above 
equation  is  true: 

P.AF=  W.BF, 


198  STATICS.  [187. 

187.  Three  Kinds  of  Lever.  The  three  forms  in  Figs. 
126,  127,  128  are  sometimes  called  the  three  kinds  of 
lever,  though  there  is  no  essential  difference  between 
them.  In  the  first  kind  the  fulcrum  is  between  the 
power  and  weight;  if  nearer  to  the  latter,  there  is  a 
mechanical  advantage;  if  nearer  to  the  power,  a  mechan- 
ical disadvantage.  If  the  arms  are  equal,  then  P  =  TT^ 
as  in  the  ordinary  balance  (191). 

In  the  second  kind  the  fulcrum  is  at  the  end,  and  the 
weight  nearer  to  it  than  is  the  power;  in  this  case  there 
is  always  a  mechanical  advantage. 

In  the  third  kind  the  fulcrum  is  at  the  end,  but  the 
power  is  nearer  to  it  than  the  weight,  and  there  is  there- 
fore a  mechanical  disadvantage. 

188.  The  first  form  of  lever  is  illustrated  by  the  crow- 
bar, by  means  of  which,  owing  to  the  great  difference  in 
the  lengths  of  the  arms,  a  very  great  resistance  can  be 
overcome.  Scissors  and  nippers  are  double  levers  of  this 
class,  and  the  handle  and  claw  of  a  hammer  form  a 
curved  lever. 

The  distinction  between  the  gain  of  power  and  loss  of 
velocity,  and  the  converse,  as  determined  by  the  position 
of  the  fulcrum,  is  illustrated  by  the  shears  used  by  a 
tinman  and  a  tailor  respectively.  Those  of  the  former 
have  short  blades  and  long  handles,  and  can  consequently 
overcome  a  great  resistance  slowly;  those  of  the  tailor 
have  short  handles  and  long  blades,  and  move  quickly, 
so  as  to  cut  yielding  materials. 

An  example  of  a  lever  of  the  second  class  is  a  wheel- 
barrow :  the  fulcrum  is  at  the  centre,  or  axis,  of  the 
wheel,  the  weight  acts  down  at  the  centre  of  gravity  of 
the  load  and  barrow  together,  and  the  power  is  applied 
at  the  handles.  A  nut-cracker  or  a  lemon-squeezer  is 
an  example  of  a  double  lever  of  this  kind. 


189.] 


LEVER. 


199 


The  human  fore-arm  is  an  example  of  a  lever  of  the 
third  class:  the  elbow-joint  is  the  fulcrum,  the  weight  is 
grasped  in  the  hand,  and  the  power  is  applied  by  a 
tendon  from  the  muscle  above  attached  very  near  the 
elbow,  and  acting  obliquely.  There  is  consequently  a 
very  serious  mechanical  disadvantage,  but  in  its  place  is 
gained  great  rapidity  of  movement.  A  pair  of  tongs  is 
another  example  of  a  lever  of  this  kind. 

189.  The  following  cases  involve  the  principle  of  the  lever. 
DF{Fig.  133  or  134)  is  a  heavy  rod  hinged  at  F,  so  that  it  is  free 
to  turn  in  a  vertical  plaae,  and  supported  either  by  a  string  carried 
from  C  up  to  JEJ  (Fig.  133),  or  by  a  prop  from  G  io  E  below  (Fig. 


Fia.  133. 


Fio.  134. 


134).  In  this  case  the  power  (P)  is  the  tension  of  the  string  (or 
thrust  of  the  prop),  and  its  moment  is  P.AF;  the  weight  is  that  of 
the  bar  acting  at  its  centre  of  gravity  B,  and  its  moment  isTT.^^. 
The  value  of  P,  derived  from  the  equation 


RAF=  W.BF, 


will  give  the  tension  of  the  string  (or  thrust  of  the  prop)  needed 
just  to  support  the  rod. 


200  STATICS.  [190. 

190.  The  Lever  on  the  Principle  of  Work.  It  has  been 
shown  in  Art.  179  that,  in  the  case  of  every  machine,  if 
friction  and  all  other  hnrtf ul  resistances  are  eliminated, 


P,s 

= 

W.h, 

p 

W 

= 

h 

s' 

W  -   T-  (1) 


Here  s  is  the  distance  through  which  the  power  acts, 
and  h  the  distance  through  which  the  weight  is  raised. 
If  a  resistance  (B)  is  overcome  through  a  distance  d, 
then 

P.s  =  B.d, 

-^  ==  -.  (2) 

By  the  use  of  these  equations  the  relation  of  tlie  Power 
to  the  Weight  (or  Kesistance),  when  the  Power  raises 
the  Weight  uniformly,  can  be  obtained.  This  value  of 
p 

-^  is  the  same  as  that  deduced  on  condition  of  equilib- 
rium, on  the  principles  of  statics. 

In  the  case  of  the  lever,  suppose  (1)  that  the  power 


FiQ.  135. 


always  acts  vertically  (Fig.  135)   and  turns  the  lever 
from  the  position  AB  to  that  of  A'B^;  then  the  effective 


190.]  LEVEE.  201 

distance  through  which  it  acts  is  ^'6'  (=  s),  and  the 
height  through  which  the  weight  is  raised  is  B^D  {=  h). 
Therefore 

_P  _    B^  _  B^  _  BF  ^ 

W  ~  A'C  ~  A'F  ~  AF' 

.-.  P.AF  =  W.BF,         as  in  Art.  186. 

(2)  Suppose  the  power  and  resistance  to  act  continu-. 
ally  at  right  angles  to  the  straight  lever  AB  (Fig.  136) 


Fig.  136. 

while  it  turns  from  the  position  ^^  to  A^B\  Here  the 
power  acts  through  the  arc  AA^  (=  s),  and  the  resist- 
ance is  overcome  through  the  distance  of  the  arc  BB' 
(=  d).     Hence 

P_  _  BB^  _  BF 
R   ""  AA'  ~"  AF' 

.\  F.AF=  E.BF 

In  general,  the  relation  may  be  obtained  after  the  same 
manner,  whatever  the  shape  of  the  lever  or  the  directions 
of  F  and  W  (or  E).  In  each  case,  however,  it  must  be 
remembered  that  s  and  h  (or  d)  are  not  necessarily  the 
actual,  but  always  the  effective,  distances. 


202  STATICS.  [191. 


B,  Some  Special  Applications  of  the  Principle  of  the 
Lever. 

I.    BALANCE. 

191.  The  Balance  is  a  contrivance  used  for  measur- 
ing the  mass  of  bodies  ;  or,  in  familiar  language,  of  deter- 
mining their  weight  by  comparison  with  that  of  certain 
assumed  units.  (See  Arts.  54,  55.)  In  its  ordinary  form 
it  consists  of  a  beam,  so  constructed  as  to  be  at  once  strong, 
rigid,  and  light.  This  beam  is  poised  on  a  knife-edge,  in 
the  middle,  as  a  fulcrum,  often  resting  on  a  plate  of  agate. 
From  the  extremities  of  the  two  equal  arms  are  sus- 
pended pans  of  the  same  size  and  weight.  The  object 
weighed  is  placed  in  one  pan,  and  the  counterpoise  is 
adjusted  to  balance  it  in  the  other. 

192.  A  good  balance  must  satisfy  these  three  condi- 
tions: it  must  be  (1)  true,  (2)  stable,  and  (3)  sensible. 

(1)  It  is  true  when  the  arms  are  of  exactly  the  same 
length  and  weight,  and  when  the  scale-pans  are  also  just 
equal.  It  will  then  be  rigidly  true  that  P  =  W.  If, 
however,  the  arm  of  the  pan  in  which  the  object  is 
weighed  is  longer,  then  a  smaller  amount  of  it  will  bal- 
ance the  given  counterpoise,  and  the  purchaser  in  such  a 
case  would  be  defrauded,  and  conversely.  This  inequali- 
ty would  be  proved  by  exchanging  the  two  objects. 
If  the  apparent  weight  in  one  pan  is  a,  and  in  the 
other  b,  the  true  weight  will  be  equal  to  Vab. 

(2)  The  balance  must  also  be  stable;  that  is,  after  being 
slightly  disturbed  it  must  return  to  its  original  position. 
In  order  to  satisfy  this  condition  the  centre  of  gravity 
must  be  below  the  axis  on  which  the  beam  turns,  for  if 
above  it  would  be  in  unstable  equilibrium,  and  if  on  the 


193.] 


BALANCE. 


203 


axis  the  equilibrium  would  be  neutral;  that  is,  the  beam 
would  balance  in  every  position  (175). 

(3)  The  balance  must  be  sensible ;  that  is,  when  the 
weight  in  one  pan  slightly  exceeds  that  in  the  other, 
this  difference  must  be  indicated  by  the  inclined  posi- 
tion of  the  beam  when  it  comes  to  a  state  of  rest.  The 
sensibility  is  obviously  greater  as  the  angle  of  deflection 
increases  for  a  constant  difference  of  weight;  this  angle 
is  often  measured  by  a  long  slender  rod  which  is  at 
right  angles  to  the  beam  and  turns  with  it.  The  degree 
of  sensibility  to  be  attained  in  a  given  case  depends 
upon  the  object  for  which  the  balance  is  to  be  used;  for 
example,  a  balance  suitable  for  use  in  chemical  analyses 
should  indicate  distinctly  a  difference  of  ^  of  a  milli- 
gram; that  is,  Tiro-inr  of  a  gram. 

193.  The  conditions  upon  which  the  sensibility  of  a 
balance  depends  are:   (1)  the  length  of  beam,  (2)  the 


^1 ^^ 

■f^- — *\ 

\ 


Fio.  187. 


weight  of  the  beam,  and  (3)  the  position  of  the  centre 
of  gravity. 

Let  A  and  B  (Fig.  137)  be  the  points  of  the  beam  at 
which  the  scale-pans  are  hung,  and  let  C  be  the  axis  or 


204  STATICS.  •  [193. 

fulcrum;  then  the  line  A  CB  will  join  these  three  points. 
Suppose  also  that  the  shape  of  beam  is  such  that  its  cen- 
tre of  gravity  is  at  G,  at  which  point  its  weight  ( Q)  con- 
sequently acts.  A'B'  (Fig.  137)  represents  the  inclined 
position  which  the  line  AB  taiies  for  a  given  difference 
of  weight  in  the  two  pans  of  W—W\  The  angle  of 
deflection  of  the  rod  FCF'  {=  «)  is  the  same  as  that 
of  the  beam.  It  is  obvious  that,  for  a  given  value  of 
W  —W,  the  greater  the  angle  a  the  greater  the  sensi- 
bility of  the  balance.  Suppose  the  whole  in  equilib- 
rium; then,  by  taking  the  moments  about  G  (156), 

W.A'K=^W'.B'H-\-Q,DQ. 

But  since  A'K'  =  B'H, 

(W-W').A'K=Q.DG, 

From  the  similar  triangles  GA'K,  DGE, 

A'K        DG  A'K        A'C 

A'G         GE'  DG     ~  GE' 

.-.     (W-W')AG=Q.GE, 

But  GE  —  CG  tan  a-,  hence 

{W -W)  AC  =  Q,CG  i2^n  a, 

(W-W).AC 
^"^"^         Q.GG       ' 

In  this  final  equation  A  O  is  one  half  the  length  of  the 
beam,  and  CG  is  the  distance  of  its  centre  of  gravity 
below  the  axis.  Now,  as  has  been  stated,  the  sensibility 
of  the  balance  increases  as  a  increases,  for  a  given  value 
( W—  Pf ').  It  is  obvious,  from  this  equation,  that  tan  a 
is  increased  (1)  by  making  AC,  the  length  of  the  beam, 


194.]  STEELYAED.  205 

greater;  also,  (2)  by  diminishing  Q,  the  weight  of  the 
beam;  and  finally,  (3)  by  diminishing  CG — that  is,  by 
bringing  the  centre  of  gravity  as  near  as  practicable  to 
the  axis. 

The  most  satisfactory  result  will  be  obtained  by  con- 
Bidering  these  conditions  together,  since  they  depend 
upon  one  another.  For  example,  if  the  length  of  the 
beam  is  increased,  its  weight  must  be  also,  in  order  that 
it  still  be  rigid;  again,  although  the  sensibility  increases 
as  the  distance  of  the  centre  of  gravity  below  the  axis  is 
diminished,  the  motion  of  the  beam,  as  it  tends  to  come 
to  a  position  of  equilibrium,  becomes  more  slow,  so  that 
there  is  also  a  practical  limit  in  this  direction. 

The  equation  shows  that  the  difference  in  weight  is 
proportional  to  tan  a,  and  for  very  small  angles  it  is 
proportional  to  the  angle  itself  (a), 

II.    STEELYARD. 

194.  Common  Steelyard.  In  the  Steelyard  we  have, 
in  the  place  of  the  fixed  arm  and  varying  counterpoise 
of  the  ordinary  balance,  a  varying  lever-arm  and  a  fixed 
counterpoise.  The  bar  is  made  heavier  at  one  extremity, 
and  to  this  end  is  attached  the  hook  or  scale-pan;  near 
it  is  the  point  of  support.  This  axis  is  consequently 
near  the  centre  of  gravity  of  the  whole,  but  usually  does 
not  coincide  with  it.  In  order  to  graduate  the  steelyard, 
it  is  necessary  to  determine  first  the  zero-point  of  the 
scale,  and  then  the  distance  to  be  marked  off  from  it  for 
each  unit  of  weight  {e.g.,  1  lb.)  and  fraction  of  it. 

Let  AB  (Fig.  138)  be  the  steelyard,  supported  at  C. 
Represent  the  weight  of  the  whole  by  Q  acting  at  the 
centre  of  gravity  G.  In  order  that  the  bar  should 
balance  horizontally  about  C  when  there  is  no  weight 


206 


STATICS. 


[194. 


on  the  hook,  it  is  necessary  to  place  the  selected  counter- 
poise P  at  such  a  point,  i>,  that 

Q.CG  =  P.CD. 

This  point  D  is  then  the  zero  of  the  scale,  or  the  posi- 
tion of  P  for  0  lbs.  at  A, 


J?  c  (^ 


<^P 


Gi-I> 


¥ia.  138. 


Fig.  139. 


Let  now  a  weight  W  be  placed  on  the  hook  so  that 
it  acts  through  A;  then  the  counterpoise  P  will  balance 
it  at  B,  if  (156)  the  moments  about  C  vanish;  that  is. 


W.AC-{-  Q.GC=  P.CB; 
or,  since  Q.GG  =  P.CD, 

W.AG=  P.CB  -  P.CD  =  P(CB 
or  W,AG=P.DB, 

W.AG 


CD), 


and 


DB 


If  If  =  1  lb.,  then  the  value  of  DB  gives  the  position 
of  the  one-pound  notch  on  the  scale,  and  at  twice  this 
distance  from  D  will  be  the  two-pound  notch,  and  so  on. 
If,  as  in  Fig.  139,  the  centre  of  gi-avity  is  on  the  other 
side  of  the  fulcrum,  the  position  of  the  zero-point  D 
will  also  be  changed,  but  the  value  of  DB  is  obtained  in 
essentially  the  same  way. 


195.] 


STEELYAED. 


207 


195.  A  form  of  the  steelyard  as  actually  employed  is 
seen  in  Fig.  140.  It  will  be  observed  that  both  sides  of 
the  bar  are  graduated,  and  moreover  there  is  a  second 


■  '  ■   ■  ■  I   ■   ■   '\l^^!  ■    ■ 


Fig.  140 

ring  to  support  it  at  C\  When  the  steelyard  is  turned 
over  and  supported  at  C",  the  weight  has  a  shorter  lever- 
arm,  and  consequently,  the  counterpoise  remaining  the 
same,  with  this  second  gi-aduation  the 
instrument  is  adapted  for  heavier  weights 
than  in  the  first  case. 

A  common  form  of  the  steelyard  is  also 
seen  in  the  post-office  scales,  where  the 
object  to  be  weighed  is  placed  on  a  plat- 
form, and  the  counterpoise  slides  along 
the  graduated  arm. 

Another  very  simple  form  of  balance 
involving  the  same  idea  of  a  varying 
lever-arm  is  seen  in  the  contrivance  often 
employed  for  weighing  letters  (Fig.  141). 
When  there  is  no  weight  at  B,  the 
weight  of  the  instrument  acts  through   its  centre  of 


Fig.  141. 


208  STATICS.  [196. 

gravity  directly  below  the  point  of  suspension  C.  If 
now  a  letter  is  placed  between  the  springs  at  B  the 
position  is  slightly  changed,  so  that  the  moment  of  its 
weight  is  equal  to  the  moment  of  the  weight  of  the 
instrument  in  its  new  position. 

196.  Danish  Steelyard.  In  the  Danish  steelyard  no 
counterpoise  is  employed,  but  the  adjustment  is  made 
by  shifting  the  position  of  the  supporting-hook,  and 
consequently  giving  the  weight  of  the  bar  a  longer  or 
shorter  lever-arm.      Let  (Fig.  142)  AB  represent  the 


I 


<l- 


Fio.  142. 


bar,  heavier  at  the  end  B,  and  let  its  weight  Q  act  at  O, 
Suppose  a  weight  W io  be  hung  on  the  hook  at  A-y  then, 
in  case  of  equilibrium,  we  have 

Q,CG=  W.AC; 
but  00  =  AG -AC, 

.-.   Q{AO^AC)=W.AO,  ot{Q-{-W)AO=Q.AO; 

,.A0-^'^^- 


Q+w 


The  arm  is  graduated  by  letting  W  =  1  lb.,  3  lbs.,  etc., 
in  succession;  since  Q  anAAO  have  constant  values,  AG 
is  thus  obtained  for  each  case. 

197.  Roberval's  Balance.     In  many  forms  of  balance 
in  common  use,  instead  of  two  scale-pans  suspended 


196.] 


STEELYAED. 


209 


M 


Fig.  142a. 


from  a  beam  above,  there  are  two  platforms  supported 
from  beneath,  upon  one  of  which  is  placed  the  object  to 
be  weighed,  and  upon  the  other  the  counterpoise;  or  (as 
in  the  post-office  scales  alluded  to  in  Art.  195)  there  is 
one  platform,  and  the  place  of  the  other  is  taken  by  a 
graduated  arm  upon  which  slides  a  constant  counter- 
poise. In  such  balances  it  is  p  p 
essential  that  the  indications  _ 
should  be  accurate,  no  mat. 
ter  what  the  position  of  the 
load  on  the  platform. 

The  way  in  which  this  end 
is  often  attained  is  illustrated 
by  RolervaVs  hdlance  (Fig. 
142«) ;  CD,  EF  are  here  two  bars  of  equal  length,  piv- 
oted to  the  upright  support  at  A  and  B\  they  are  also 
jointed  at  (7,  E,  and  D,  F,  thus  forming  a  rectangular 
frame.  The  equal  pans  are  supported  at  H  and  K, 
and  upon  them  respectively  are  placed  the  object  to  be 
weighed  and  the  counterpose,  as  P,  P  at  if  and  N.  (In 
actual  use  the  frame  CDFE  is  generally  concealed  in  the 
stand  of  the  balance. ) 

In  the  figure  it  is  seen  that  the  weights  P,  P  are  at 
very  unequal  distances  from  the  axis  AB,  but  the  ac- 
curacy is  not  impaired  by  this  fact.  To  prove  this,  sup- 
pose two  opposite  forces,  each  equal  to  P,  to  act  at  K, 
and  two  others  similar  at  H;  they  will  not  alter  the 
previous  conditions.  We  have  now,  in  place  of  P  at  N, 
a  force  equal  to  P  acting  downward  at  K,  and  a  couple 
(150)  whose  moment  is  P,KN;  also,  in  place  of  P  at 
M,  we  have  an  equal  force  acting  downward  at  H,  and  a 
couple  whose  moment  is  P.HM.  The  two  equal  down- 
ward forces  at  H  and  K  will  obviously  balance  each 


210 


STATICS. 


other;  the  couples  though  unequal  do  not  disturb  the 
equilibrium,  for  they  merely  produce  unequal  strains  at 
the  fixed  points  A  and  B,  and  do  not  alter  the  effect  of 
the  other  forces. 


III.    TOGGLE-JOINT. 

198.  Toggle- Joint.  Fig.  143  represents  two  combined 
levers,  AB,  BO,  forming  what  is  called  a  toggle- 
joint.  They  are  hinged  together  at  B,  forming  an 
angle  ABC=2a  ;  further,  the  lever  AB  turns  freely  at 
Ay  while  the  end  0  of  BG  is  free  to  move  in  the  direc- 
tion X2/,  and  acts  against  the  resistance  Q.     Suppose  the 


Fio.  143. 


force  P  to  act  vertically  downward  at  B  against  the  resis- 
tances R  and  R;  if  the  system  is  in  equilibrium,  we  have, 
by  Art.  133, 


F  _  sin  ABC 
R  ~  sin  FBC 


sin  2a       2  sin  a  co^  a 


sm  a 


sm  a 


=  2  cos  a. 


But  the  effective  resistance  Q  is  only  one  component  of 
R,  the  other  being  supplied  by  the  reaction  of  the  plane 
xi/;  hence 

Q=:R  sin  a,         —^  t>  -     Q 


P 

Q 


and 

2  cos  rtr  _ 
sin  a 


R 
2 


sm  a 


tan  a 


199.] 


TOGGLE-JOINT. 


211 


As  now  the  levers  AB,  BC  straighten  out,  the  angle 
2a  approaches  180°,  and  tan  a  approaches  infinity  as 
its  limit;  the  mechanical  advantage,  therefore,  when 
the  levers  are  nearly  in  a  straight  line  is  enormously 
great,  and  hence  a  very  great  resistance  can  be  over- 
come. The  toggle-joint  is  seen  in  the  arrangement  by 
which  the  cover  of  a  carriage  is  raised. 

199.  The  principle  of  tlie  toggle-joint  and  its  great  efficiency 
are  well  illustrated  by  the  Stone- Crusher  invented  by  Mr.  Blake, 
of  New  Haven.     The  accompanying  cut  (Fig.  144)  gives  a  longi- 


FiQ.  144. 


tudinal  section  of  the  machine.  The  parts  show  so  clearly  the 
relations  explained  in  the  preceding  article  that  but  little  descrip- 
tion is  needed.  The  power  is  applied  to  the  wheel  B,  and  as  it 
revolves  the  central  post  alternately  rises  and  falls.  This  serves 
to  work  the  levers,  or  toggles,  0,  0.  One  end  of  them  is  station- 
ary on  the  right  (corresponding  to  A  in  Fig.  143),  and  the  other 
acts  on  the  movable  iron  mass  J  swung  on  K.     Finally  this  re- 


212  STATICS.  [200. 

suits  in  the  opening  and  shutting  of  the  jaws  P,  P.  The  distance 
through  which  the  movable  jaw  works  is  small,  and  the  force 
exerted  against  any  object  placed  between  is  enormous.  In  use 
the  blocks  of  stone  or  ore  are  fed  in  from  above,  and  the  motion 
of  the  jaws  rapidly  crushes  them  down  to  uniform  fragments  of  a 
size  regulated  by  the  distance  between  the  jaws  at  the  lowest  points ; 
the  fragments  pass  out  by  the  shute  A.  Such  a  machine  will 
yield  about  10  tons  of  broken  rock  per  hour. 

rV.    COMPOUlSfD   LEVERS. 

200.  It  is  seen  in  Art.  186  that  in  the  lever,  by  making 
the  arm  of  the  weight  very  short  and  that  of  the  power 
very  long,  any  required  mechanical  advantage  may 
theoretically  be  obtained.  In  j)ractice,  however,  vari- 
ous difficulties  would  obviously  arise  in  an  attempt  to 
gain  power  in  this  way.  To  avoid  them,  and  at  the 
same  time  to  have  greater  compactness,  it  is  found 
better  to  employ  a  series  of  levers,  in  which  the  Aveighfc 
of  the  first  becomes  the  power  of  the  second,  and  so  on. 

In  Fig.  145,  let  AC,  DF,  GK  be  three  levers,  ar- 
ranged as  just  indicated.     Q  is  at  once  the  weight  of  the 


Fm.  145. 

first  and  the  power  of  the  second  lever,  and  the  same  is 
true  for  8  with  reference  to  the  second  and  third  levers. 
Now  if  the  whole  is  in  equilibrium: 

P__BG^  Q_EF^  S  _HK 

Q  ~  BA'         S~  ~  UD'  W  ~  HG' 

By  multiplying  these  ratios  together, 

P^_  B0_       EF_      HK^ 

W  BA^  ED^  HG' 


201.]  COMPOUND  LEVERS.  213 

It  is  seen  here  that 

The  final  mechanical  advantage  is  equal  to  the  product 
of  the  several  values  for  each  successive  lever.  This 
principle  is  true  not  only  for  levers  in  combination,  but 
in  general  for  any  compound  machine. 

p 

In  determining  the  ratio  of  -rr^  for  a  compound  ma- 
chine on  the  principle  of  work,  it  is  to  be  noted  that  the 
essential  point  is  to  know  the  distances  through  which 
P  and  W  act.  If  these  can  be  determined,  their  in- 
verse ratio  gives  the  ratio  required,  and  the  intermediate 
steps  in  the  machine  are  of  no  importance. 

201.  The  principle  of  the  compound  levers  finds  an 
important  application  in  the  scales  used  for  weighing 
very  heavy  objects,  as,  for  example,  hay-scales,  or  rail- 
road scales  for  cars  loaded  with  coal  (say  10  tons  each). 
By  a  combination  of  a  series  of  levers,  and  at  the  same 
time  by  the  suitable  distribution  of  the  weight  brought 
on  the  platform,  this  being  supported  at  a  number  of 
points,  very  heavy  weights  may  be  determined  with  all 
desirable  accuracy.  The  whole  is  balanced  by  a  counter- 
poise, or  series  of  them,  used  in  connection  with  a  steel- 
yard arm. 

EXAMPLES. 

XXVI.  Lever.    Articles  185-190. 

[The  weight  of  the  lever  is  to  be  neglected,  except  when  otherwise 
stated.  1 

.    1.  The  force  P=  40  lbs.  acts  as  in  Fig.  126,  p.  196;  AF  = 
8  feet  and  AB  =10  feet:  What  weight  can  be  supported  ? 

2.  If  (Fig.  127,  p.  196)  AB  =  10,  BF  =  2,  and  the  weight  is 
120  lbs.,  what  force  P  is  required  to  support  it  ? 

3.  If  (Fig.  128,  p.  196)  AB  =  14,  AF  z=  2,  and  P  =  100  lbs., 
what  weight  can  P  support? 


214  STATICS.  [201* 

4.  What  is  the  pressure  on  the  fulcrum  in  each  of  the  above 
cases  ? 

5.  AFG  is  a  bent  lever  (Fig.  129,  p.  197);  AF  =  14,  FC  =  16, 
AFC  =  135°,  P  =  30  lbs. :  What  is  TT? 

6.  CFJ)  is  a  bent  lever  (Fig.  131,  p.  197);  CF  =  18,  FD  =  13, 
PDF  =  150°,  FCW  =  165°,  and  TT  =  60  lbs. :  What  is  P  ? 

7.  If  in  Fig.  130,  p.  197,  CB  =  16,  BF  =  2,  and  ACF=  80°, 
also  P  =  40  lbs.,  what  is  W ? 

8.  What  is  the  pressure  on  the  fulcrum  in  examples  5  and  7  ? 

9.  A  heavy  uniform  rod  Z>i^(Fig.  133,  p.  199),  weighing  25  lbs 
and  2  feet  long,  is  hinged  at  F;  it  is  supported  by  a  string  carried 
from  G{GF  —  20  in.)  to  a  point  E,  12  inches  vertically  above  F\ 
What  is  the  tension  of  the  string  ? 

10.  A  heavy  uniform  shelf  DF  (^\g.  134),  18  in.  wide  (=  BF), 
weighing  36  lbs. ,  and  hinged  at  F,  is  supported  by  a  prop  carried 
from  G  {GF=  12  in.)  to  a  point  ^  below  F,  so  that  GF  =  FE: 
What  pressure  does  this  prop  feel  ? 

11.  A  uniform  stick  8  feet  long,  weighing  2  lbs. ,  is  supported 
between  the  thumb  and  first  finger ;  the  one  acts  at  the  extremity 
as  a  fulcrum,  and  the  other  as  a  force  at  right  angles  an  inch  from 
it :  What  is  the  force  required  when  the  stick  is  horizontal  ?  when 
inclined  60°  to  the  horizontal  ? 

12.  A  rod  weighing  10  lbs.  has  a  weight  of  10  lbs.  at  one  end 
and  of  20  lbs.  at  the  other :  Where  must  the  fulcrum  be  in  case  of 
equilibrium  ? 

13.  Forces  of  8  and  12  lbs.  act  at  the  extremities  of  a  bar  16  feet 
long,  and  in  directions  making  angles  of  135°  and  150°  respectively 
with  it :  Where  is  the  fulcrum  in  case  of  equilibrium  ? 

XXVII.  Balance,    Articles  191-193. 

1.  A  body  is  equivalent  to  a  weight  of  12  lbs.  in  one  pan  of  a 
false  balance,  and  of  16^  lbs.  in  the  other:  What  is  the  true 
weight  ? 

2.  A  body  is  equivalent  to  a  weight  of  6  lbs.  4  oz.  from  one  arm 
of  a  false  balance,  and  of  4  lbs.  6  oz.  from  the  other :  What  is  the 
ratio  of  the  lengths  of  the  arms  ? 

3.  The  true  weight  of  a  body  is  15  oz.,  its  apparent  weight  in 
one  pan  of  a  balance  is  1  lb. :  What  would  it  seem  to  weigh  in 
the  other  pan  ? 


WHEEL  AKD  AXLE.  215 


XXVIII.  steelyard.    Articles  194-196. 
[The  common  steelyard  (194)  is  intended  unless  otherwise  stated.] 

1.  The  longer  arm  of  a  steelyard  is  26  inches  in  length,  the 
shorter  2f  inches;  the  arrangement  of  .the  scale-pan  (or  hook)  is 
such  that  P(=  2  lbs.)  at  B  (Fig.  139,  p.  206),  if  BC  =  10  inches, 
balances  8  lbs.  at  A :  (a)  Where  is  the  zero  of  the  scale  ?  (b)  If  the 
whole  weighs  1^  lbs.  (=  Q),  where  is  the  centre  of  gravity  ?  (c)  What 
must  be  the  graduation  for  ounces  ?  (d)It  P  cannot  be  conveni- 
ently brought  nearer  than  f  of  an  inch  to  G,  what  are  the  greatest 
and  least  weights  for  which  it  can  be  used? 

2.  The  whole  length  of  a  steelyard  is  24 inches;  CG  (Fig.  138)  = 
J  in.,  CA  =  ^  in.,  P=8  oz.,  and  ^  =  1  lb. :  (a)  Where  is  the 
zero  of  the  scale  ?  (b)  What  is  the  length  of  graduation  for  1  lb.  ? 
(c)  How  large  weights  can  it  be  used  for  ? 

3.  In  Fig.  140,  p.  207,  ^C=  3  in.  and  ^C"  =  1  in. ;  the  counter- 
poise =  12  oz. :  What  is  the  length  of  a  division  on  the  scale  for 
1  oz.  in  each  position  ? 

4.  The  weight  of  the  beam  of  a  steelyard  is  3  lbs.,  and  the 
distance  of  its  centre  of  gravity  is  i  inch  from  the  fulcrum :  Where 
must  a  counterpoise  of  1  lb.  12  oz.  be  placed  to  balance  it  ? 

5.  The  length  of  a  Danish  steelyard  is  30  in.,  its  weight  is 
4  lbs.,  and  acts  at  a  point  3  in.  from  one  end;  a  body  weighing 
12  lbs.  hangs  at  the  other  end :  Where  is  the  fulcrum  ? 

6.  The  length  of  a  Danish  steelyard  is  28  in.,  its  weight  is 
3  lbs.,  acting  at  a  point  4  in.  from  one  end:  (a)  Where  is  the  l-lb 
notch?(6)  the2-lb.? 

II.  Wheel  and  Axle. 

202.  The  Wheel  and  Axle,  in  its  simplest  form, 
consists  of  two  cylinders  of  different  sizes,  rigidly  con- 
nected and  turning  about  a  common  axis;  the  larger  is 
called  the  wheel,  and  the  smaller  the  axle.  The  power 
is  applied  to  the  end  of  the  rope  wound  about  the  wheel, 
and  the  weight  is  raised  by  a  rope  wound  upon  the  axle. 
This  is  seen  in  Fig.  146. 


216 


STATICS. 


203.  Suppose  the  power  and  weight  to  act  in  the  same 
plane  perpendicular  to  the  axis,  as  in  Fig.  147.  Treated 
in  this  way,  it  is  essentially  a  form  of  lever,  the  fulcrum 
being  at  F\  and  in  case  of  equilibrium  the  condition 
(156)  will  hold  good  that  the  algebraic  sum  of  the  mo- 


Fio.  146. 


Fia.  147. 


ments  of  P  and  W  about  the  axis  must  be  equal  to  zero. 

That  is, 

P.AF=  W.BF. 


If  AF  =  R,  and  BF  =  r,  then 

r 


P        r 

=  »-;   or- 


The  Power  is  to  the  Weight  as  the  radius  of  the  axle  is 
to  the  radius  of  the  wheel. 

204.  Since  the  power,  if  applied  by  a  rope  as  here, 
must  always  act  at  right  angles  to  the  radius  of  the 
wheel,  the  relation  given  above  holds  good  whatever  the 
direction  of  P.  The  pressure  upon  the  axle,  however, 
will  vary:  if  P  and  W  are  parallel  and  in  the  same  direc- 
tion, it  is  equal  to  their  sum;  if  in  opposite  directions. 


207.]  WHEEL  AND  AXLE.  217 

to  their  difference;  and  in  other  cases  it  is  to  be  obtained 
by  the  parallelogram  of  forces. 

205.  The  thickness  of  the  rope  is  here  neglected. 
Strictly  speaking,  half  of  this  thickness  should  be  added 
to  each  of  the  radii.  As  the  weight  is  raised,  and  its 
rope  consequently  wound  up  on  the  axle,  thus  increasing 
its  radius,  and  also  that  of  the  power  is  unwound,  thus 
diminishing  its  radius,  the  relation  of  P  to  TT  will  con- 
tinually vary,  and  in  general,  for  the  case  supposed,  P 
must  increase. 

206.  The  Wheel  and  Axle  on  the  Principle  of  Work. 

Suppose  the  power  to  continue  to  act  uniformly  while 
the  system  makes  one  complete  revolution;  then  (179) 
s  =  2;ri?,  and  h  =  27tr.     Therefore 

F,27tE  =  W.27tr, 

or  F.E  =  W.r; 

P        r 
.  *.     -T^  =  -ttj      as  in  Art.  203. 

207.  Applications  of  the  Principle  of  the  Wheel  and 
Axle.  The  principle  of  the  wheel  and  axle  applies  to 
many  other  cases  besides  that  here  described.  For 
example,  the  same  relation  holds  good  for  the  common 
windlass,  as  where  a  bucket  is  raised  from  a  well,  the 
power  acting  at  the  end  of  a  crank-arm  (Fig.  148).  It 
also  applies  to  the  steering-wheel  of  a  ship,  where  the 
power  is  applied  to  the  handles  on  the  circumference; 
or  to  the  capstan  (Fig.  149),  where  the  axis  is  vertical, 
the  power  acts  on  a  handle,  and  the  rope  connected  with 
the  weight  (or  resistance)  leaves  the  axle  in  a  horizontal 
direction. 


218 


STATICS. 


[207. 


A  good  example  of  this  principle  is  seen  in  the  fusee 
(Fig.  150),  which  is  applied  to  some  watches  and  clocks. 


FiG.148. 


As  the  spring  unwinds  its  force  diminishes;  but  by  means 
of  the  fusee  it  is  made  to  act  on  a  continually  increasing 
lever-arm,  and  by  a  proper  adjustment  the  moment,  or 
turning  power  of  the  force,  can  be  kept  constant. 


Fig.  149. 


Fio.  150. 


The  wheel  of  a  yehicle  is  useful  in  reducing  frictional 
resistance,  as  explained  in  Art.  86;  in  overcoming  ob- 
stacles in  the  road  it  acts  as  a  continuous  lever,  hence 
the  advantage  of  a  large  wheel. 


208.] 


WHEEL  AND   AXLE. 


219 


208.  Chinese  Windlass.  The  combination  of  the 
wheel  and  axle  seen  in  Fig.  151  is  sometimes  called  the 
Chinese  Windlass^  or  the  differen-  ^ 

tial  windlass.  There  are  here  two  ^yJ^^AW  V 
axles  of  different  sizes,  and  the  ''^  '  J^^^  UJJf 
arrangement  is  such  that  as  the 
rope  is  wound  up  on  the  larger  axle 
it  is  unwound  on  the  smaller  one, 
it  passing  under  a  movable  pulley 
which  supports  the  weight.  The 
upward  ascent  of  the  weight  is  con- 
sequently very  slow,  but  the  me- 
chanical advantage  very  great.  Let 
R  be  the  radius  of  the  crank-arm, 
and  a  and  h  respectively  those  of  the  larger  and  smaller 
axles.  The  tension  of  the  rope  is  obviously  ^  W.  If  the 
machine  is  in  equilibrium,  then,  since  the  rope  tends  to 
turn  the  smaller  axle  in  the  same  direction,  and  the 
larger  in  the  opposite  direction,  to  the  power,  by  the 
equality  of  moments: 

F.R  =  iW{a-i), 
P       a-b 


Fio.  151. 


w 


2R 


This  result,  also  easily  obtained  by  the  principle  of 
work  (206),  shows  that  the  mechanical  advantage  be- 
comes very  great  as  the  difference  between  the  size  of 
tiie  axles  is  diminished. 


220  STATICS.  [209. 

EXAMPLES. 
XXIX.   Wheel  aTid  Axle.    Articles  20^208. 

1.  The  radius  of  the  axle  is  2  inches,  that  of  the  wheel  is  2^ 
feet,  and  the  power  acting  is  80  lbs. :  What  weight  is  supported  ? 

2.  A  horse  exerting  a  force  of  800  lbs.  walks  in  a  circle  having 
a  diameter  of  18  feet  and  turns,  by  means  of  a  lever-arm,  a  verti- 
cal post  about  which  a  rope  is  wound:  If  the  diameter  of  the  post 
is  8  inches,  what  resistance  (e.g. ,  that  of  a  building  which  is  being 
moved)  can  the  horse  overcome  ? 

3.  Four  men,  each  exerting  a  force  of  60  lbs.  acting  on  separate 
lever-arms,  4  feet  long,  turn  a  capstan ;  the  radius  of  the  circle 
about  which  the  rope  is  wound  is  6  inches :  "What  is  the  pull  felt 
upon  the  anchor? 

4.  A  weight  of  500  lbs.  hangs  by  a  rope  1  inch  in  thickness; 
r  =  8  in,,  and  R  =  4  feet;  the  power  acts  on  a  lever-arm  without 
a  rope :  What  is  P  ? 

5.  A  power  of  12  lbs.  balances  a  weight  of  200  lbs. ;  the  radius 
of  the  axle  is  3  inches :  What  is  the  diameter  of  the  wheel  ? 

6.  In  Fig.  148,  B  =  18  in.,  and  the  weight  of  250  lbs.  rises  2  feet 
while  the  power  makes  5  revolutions:  What  is  P ? 

III.  Toothed  Wheels. 

209.  A  Toothed  Wheel  is  a  circular  disc  provided 
with  teeth  on  the  circumference;  such  a  wheel  turning 


Fig.  153. 


on  one  axis  interlocks  with  a  second  turning  on  another 
axis  (Fig.  152),  and  in  this  way  the  force  applied  at 


210.] 


TOOTHED  WHEELS. 


221 


the  first  is  communicated  to  the  second.  There  may 
be  a  mechanical  advantage  with  a  corresponding  loss  of 
speed,  or  a  gain  in  velocity  and  a  consequent  mechani- 
cal disadvantage. 

When  the  wheels  are  small,  the  teeth  are  nearly  rect- 
angular in  form.  When  the  wheels  are  very  large  and 
great  force  is  employed,  the  shape  of  the  teeth  is  a 
matter  of  essential  importance,  in  order  that  the  loss  of 
power  arising  from  their  mutual  friction  and  resistance 


Fig.  153. 

may  be  a  minimum.     The  detailed  development  of  this 
subject  belongs  to  applied  mechanics. 

As  seen  in  Fig.  152,  when  the  wheels  are  turned,  the 
teeth  roll  upon  one  another  so  that  the  points  in  which 
their  mutual  resistance  is  felt,  due  to  the  power  and 
weight  acting,  lie  very  nearly  at  the  touching  points  of 
the  two  circles  drawn.  These  ideal  circles  are  called  the 
pitch-circles,  and  are  concentric  with  the  wheels  them- 
selves respectively. 

210.  Relation  of  P  to  W.  In  Fig.  153,  let  the  power 
(P)  act  on  the  radius  B;  while  the  weight  ( W),  supported 


222  STATICS.  [210. 

by  the  rope,  acts  on  the  radius  r,  they  tend  to  turn  their 
respective  wheels  in  the  direction  of  the  arrows.  The 
resistance  between  the  two  wheels  is  felt  (209)  in  the 
line  §§',  and  if  the  system  is  in  equilibrium,  by  the 
l)rinciple  of  the  lever  (156),  the  moment  of  §'  about  A 
is  equal  and  opposite  to  that  of  P,  and  of  Q  about  B  to 
that  of  W.     Hence  the  relations: 

P.R=  Q'.AG, 

and  W,r  =  Q.BC; 

P.R  _  Q\AO^ 


that  is, 


W,r  ~    Q.BG' 

P.R  _    AC__  27rAO 
W.r  ~    BG  "  27rBG' 


But  since  the  number  of  teeth  in  the  two  wheels  is  pro- 
portional to  their  circumferences,  if  ^  =  the  number  of 
teeth  in  the  power-wheel,  and  T  those  in  the  weight- 
wheel,  we  have 

P.R  _    t_ 

W.r  ~  T' 
This  may  be  stated: 

The  moment  of  the  Power  is  to  the  moment  of  thQ 
Weight  as  the  numler  of  teeth  in  the  Power-wheel  is  to 
the  number  of  teeth  in  the  Weight-wheel. 

The  final  equation  above  may  be  written: 

W  ~  R   ^   T' 

This  is  another  application  of  the  principle  explained 
in  Art.  200,  since,  in  the  case  supposed,  the  machine  is 
really  compounded  of  the  wheel  and  axle  and  the  toothed 
wheels. 


212.] 


TOOTHED   WHEELS. 


223 


211.  Toothed  Wheels  on  the  Principle  of  Work.  Sup- 
pose (Fig.  153)  that  the  power  continues  to  act  through 
one  circumference  of  its  lc\'er-arm,  'ZttE  (=  s);  if  no 
toothed  wheels  intervened,  the  weight  would  rise  through 
a  distance  equal  to  the  circumference  of  the  axle, 
27rr  (=  h).  But  the  ratio  of  the  distances  through 
which  the  toothed  wheels  will  turn  is  the  inverse  ratio  of 
their  number  of  teeth;  that  is,  if  the  smaller  wheel  has 
20  and  the  larger  40  teeth,  the  former  will  revolve  twice 

/40\  T 

\9n  J'  ^^^^®  ^^®  other  turns  once;  or,  in  general,  — .    There- 
fore 


as  above. 


212.  Application  of  Toothed  Wheels.     An  illustration 
of  the  use  of  this  machine  is  seen  in  Fig.  154,  to  which 


Fio.  154. 


the  final  equation  of  Art.  210  applies.     Severa*  series  or 
''trains"  of   toothed  wheels  are  employed  in  derricks 


224 


STATICS. 


[213. 


and  cranes  for  raising  very  large  weights.  One  of  these, 
with  three  pairs  of  toothed  wheels,  is  represented  in 
Fig.  155.  The  relation  of  the  power  to  the  weight  here, 
by  the  preceding  principles,  is: 

t 


W  ~  R       T       T'      T" 


For  example,  if 


P  =  10  lbs.,  the  radius  of  the  axle 
inches,  that  of  the  crank-arm  {R)z=z%\  feet; 
P  if,  also,  the  number  of 
teeth  in  each  of  the 
smaller  wheels  {t,  t' ,  t") 
is  20,  and  of  the  larger 
wheels  {T,  T,  T")  120, 
then  the  weight  which 
could  be  raised,  all  hurt- 
ful resistances  being  left 
out  of  account,  would  bo 
21,600  lbs.,  or  about  10 
tons. 

213.  An  excellent  illustra- 
tion of  the  use  of  tootlied 
wheels,  where  a  "gain  In 
power"  is  required,  is  af- 
forded by  the  "back  gears" 
of  a  large  turning-lathe.  The 
arrangement  is  such  that  the 
cone-pulley  F  (Fig.  156),  by  which  the  power  is  applied,  turns 
independently  of  the  wheel  L  and  of  the  spindle  of  the  lathe, 
which  last  are  attached  together.  The  "back  gears"  are  two 
toothed  wheels  attached  to  a  common  axis,  and  so  placed  behind 
the  wheels  G  and  L,  respectively,  that  by  a  slight  adjustment  they 
can  be,  when  required,  put  in  gearing  with  O  and  L.  When  the 
simple  motion  of  the  lathe  only  is  needed,  a  pin  connects  the 
cone-pulley  F  with  L,  and  then  the  spindle  is  turned  directly  at  a 


Fig.  155. 


216.]  TOOTHED   WHEELS.  225 

rate  depending,  as  mentioned  in  Art.  216,  on  the  pulleys  over 
which  the  belt  passes.  If,  however,  a  greater  resistance  is  to  be 
overcome,  as  when  a  large  object  is  to  be  turned,  the  pin  connect- 
ing ^  and  L  is  taken  out,  and  the  "  back  gears"  connected  with  G 
and  L.  The  motion  of  the  wheel  O  is  then  communicated  to  the 
larger  wheel  behind ;  this  gives  to  the  axis  of  the  latter  a  speed 
reduced  in  the  ratio  of  the  number  of  teeth  of  the  two.  This 
axis  turns  the  second  small  wheel  behind  L,  and  this  gives  motion, 
reduced  in  rate  as  before,  to  L  and  the  spindle  connected  with  it. 
If  the  number  of  teeth  in  the  two  pairs  of  wheels  are  respectively 
20  and  80,  then  the  speed  of  the  spindle  is  reduced  by  this  con- 
trivance 16  times,  and  a  corresponding  mechanical  advantage  is 
gained. 

214.  The  rach  and  pinion  consists  of  a  straight  bar 
in  which  teeth  are  cut,  into  which  fits  the  toothed 
wheel,  which  is  revolved  by  a  handle  or  screw-head. 
By  this  means  the  bar,  and  that  to  which  it  is  attached, 
is  raised  or  depressed.  This  arrangement  is  often  em- 
ployed in  practice;  for  example,  in  moving  the  tube  of 
a  microscope  up  or  down. 

215.  Toothed  wheels  are  extensively  used  in  the 
works  of  a  clock.  The  point  practically  considered  is 
here  the  relative  velocity  to  be  given  to  the  successive 
axes;  the  relation  of  P  to  TF  is  not  taken  into  account. 

216.  Use  of  Belts.  In  machines  the  motion  of  one 
axis  is  communicated  to  another  by  the  use  of  belts,  as 
well  as  by  toothed  wheels;  there  may  or  may  not  be  a 
change  of  velocity.  The  use  of  the  belt  or  strap  depends 
on  the  friction  of  the  surfaces  in  contact  (85). 

The  velocities  of  the  two  axes,  assuming  that  the  strap 
does  not  slip  at  all,  are  in  the  inverse  ratio  of  the  radii 
of  the  wheels,  and  the  mechanical  advantage  is  in  the 
direct  ratio,  as  was  true  of  the  toothed  wheels. 

Thus,  in  the  case  of  the  cone-pulley  {E,  Fig.  156)  on 


226 


STATICS. 


[217. 


the  shaft  communicating  the  power  in  the  shop,  and 
that  {F)  of  the  lathe  below,  the  velocity  of  the  axis  of 
the  lathe  will  be  greatest,  and  the  power  of  oyercoming 


resistance  the  least,  when  the  belt  passes  over  the  largest 
wheel  of  the  latter,  and  conversely. 

In  general,  according  as  the  straps  are  or  are  not 
crossed,  the  motion  of  the  second  wheel  is  in  the  oppo- 
site or  the  same  direction  as  that  of  the  first. 

IV.  Pulley. 

217.  The  Pulley  consists  of  a  circular  wheel  turning 
about  an  axis  which  is  attached  to  a  surrounding  frame, 
called  the  block.  About  the  circumference  of  the  wheel, 
which  is  grooved,  passes  a  rope,  and  at  one  end  of  this 
the  power  acts.  Sometimes  two  wheels  are  placed  side 
by  side,  as  in  Fig.  157. 

The  necessity  of  making  the  wheel  turn  on  its  axis 
arises  from  the  friction,  which  is  very  much  diminished 
in  this  way;  except  for  this,  fixed  pegs  would  answer  as 
welL 


219.] 


PULLEY. 


227 


The  efiBciency  of  the  pulley  is  based  upon  the  prin- 
ciple (122)  that  the  tension  of  a  given  string  is  the  same 
at  every  point. 

218.  Single  Fixed  Pulley.     In  the  single  fixed  pulley 


Fig.  157. 

the  Power  is  equal  to  the  Weight.     This  relation  follows 
immediately  from  the  principle  stated  above,  for  (Fig. 
158)  the  tension  of  the  rope  on  both 
sides  of  the  wheel  A   must   be  the     3>        |    ^  "V 
same,  and  to  satisfy  this  condition  we  ^ 

must  have 

P  =  W. 


^ 


k^ 


Fig.  158. 


There  is  therefore  no  mechanical  ad-  ^/P 

vantage  in  the  use  of  the  single  fixed 
pulley,  but  it  serves  to  change  the 
direction  of  the  force  applied.  The 
tension  on  the  beam  at  a  is  equal  to  2P. 

219.  Single  Movable  Pulley  with  Parallel  Strings.    In 

the  single  movalle  pulley  ivith  parallel  strings  the  Weight 
is  twice  the  Power,  The  tension  on  both  sides  of  the 
wheel  A  (Fig.   159)  must,  as  before,  be  equal  to  P; 


228 


STATICS. 


[230. 


therefore  W  is  supported  by  two  upward  forces,  each 
equal  to  P,  and 

W=2P,        or        P  =  iW. 

t,  as  in  Fig.  160,  the  rope  passes  oyer  a  second  fixed 


Fig.  159. 


Fig.  160. 


pulley,  no  change  is  made  by  this  (218),  for  it  is  still 
true  that  W  =  2P.  The  tension  at  b  is  P,  and  at  a 
(Fig.  160)  it  is  2P. 

220.  Single  Movable  Pulley  with  Inclined  Strings.    It 

was  assumed  in  the  preceding 
article  that  both  branches  of  the 
rope  were  parallel;  if,  however, 
they  are  inclined  at  an  angle  2a 
fJP  (Fig.  161),  then 

IF  =  2P  cos  a. 

In  Fig.  161  the  tension  of  the 
rope  on  both  sides  of  the  pulley 
is,  as  before,  P,  but  here  W  is 
supported  not  by  two  forces  each 
equal  to  P,  but  only  by  their  components  acting  verti- 
cally upward.     Since  (131,  a)  the  vertical  line  bisects 


Fig.  161. 


221.1 


PULLEY. 


229 


the  angle  2a,  these  components  are  equal,  and  each  has 
the  value  P  cos  a;  hence 

W  =  2P  cos  a. 

There  is  evidently  a  mechanical  disadvantage  here  as 
compared  with  the  preceding  case,  for  W  =  2P  only 
when  a  =  0°,  the  two  strings  being  parallel,  and  P 
increases  as  a  increases.  When  a  =  60°  (2a  =  120°), 
then  P  =  W;  and  when  a  =  90°  {2a°  =  180°)  and  the  rope 
is  horizontal,  P  =  co  ;  that  is,  if  the 
rope  were  perfectly  flexible,  no  finite 
force  could  draw  it  out  horizontal. 

221.     Combinations    of    Pulleys. 

First  System.  Fig.  162  represents 
what  is  called  the  first  system  of 
pulleys ;  here  W  =  2"P.  The  pul- 
ley A  is  supported  by  two  forces, 
each  equal  to  P;  these  act  on  the 
string  which  passes  under  the  pulley 
B,  so  that  its  tension  is  2P.  The 
pulley  B  is  consequently  supported 
by  two  forces,  each  equal  to  2P. 
Again,  the  tension  of  the  string  pass- 
ing from  B  under  G  is  2^P;  and  C  is  supported  by  two 
equal  forces,  each  equal  to  2^P;  similarly  for  D;  and 
finally,  the  pulley  JEJ  is  acted  upon  by  two  upward  forces, 
each  equal  to  2*P.     Hence 


Fig.  162. 


or,  in  general, 


W  =  2'P, 
W  =  2'*P, 


where  n  is  the  number  of  the  pulleys. 


230 


STATICS. 


The  beam  supports  at  a  a  tension  cl  F,  also  2P  at  b, 
4.P  at  c,  8P  at  d,  and  16P  at  e. 

222.  Second  System.  W  =  nP.  Fig. 
163  represents  another  system  of  pul- 
leys. As  here  but  one  string  is  involved, 
its  tension  throughout — that  is,  each  of 
the  six  branches — is  equal  to  P.  The 
weight  is  therefore  supported  by 
forces,  each  equal  to  P;  that  is. 


SIX 


or,  in  general, 


Tr=  QP, 


W  =  nP, 


where  n  is  the  number  of  strings  rising 
from  the  movable  pulleys. 

This  form  of  pulleys  is  the  one  which 
is  generally  employed,  though  in  practice, 
as  remarked  in  Art.  226,  the  wheels  are 
placed  side  by  side  in  two  blocks. 

If  in  this  arrangement  the 
weight  w  of  the  movable  block  is 
taken  into  account,  the  relation  is 
then 

W-\-w  =  nP, 

223.  Third  System.  In  the  sys- 
tem of  pulleys  shown  in  Fig.  164, 
W=^  (2«-l)P. 

The  tension  of  the  string  on 
which  the  power  acts,  that  is  at  a^ 
is  P;  hence  the  pull  on  the  wheel 
B  ia2P,  and  this  force  is  felt  up- 
ward at  b'y  still  again,  the  pull  on  C  is  4P,  or  2'P,  and 


Fig.  164. 


.] 


PULLEY. 


231 


the  same  upward  at  c;  also,  on  D  it  is  8P,  and  the 
same  upward  at  d.  That  is,  the  weight  is  supported  by 
the  four  forces ;  yiz.,  P  +  2P  +  2'P  +  2'F  =  UP. 
Hence 

W=15P=  {2'  -  1)P, 

or,  in  general, 

r  =  (2«  -  1)P. 
The  tension  on  the  beam  is  equal  to  W  -\-  P,  or  2* P. 

224.  The  following  are  other  forms  of  pulleys,  for 
which  the  relations  of  P  to  TT  can  be  established  in  the 
same  manner  as  for  those  already  explained. 

In  Fig.  165,  W  =  4.P.  In  Fig.  166,  W  =  6P  cos  a 
(the  tendency  of  the  horizontal  component  of  P  is  also  to 
be  noted).     In  Fig.  167,  ^  =  81P  =  3*P. 


Fig.  167. 


225.  The  Pulley  on  the  Principle  of  Work.  1.  Single 
Fixed  Pulley,  Here  (Fig.  168)  the  power  and  weight 
act  through  equal  distances;  that  is,  s  =  h,  and  there- 
fore W  =  P, 

2,  Single  Movable  Pulley  with  Parallel  Strings.     In 


232 


STATICS. 


[175. 


this  case  (Figs.  169,  170)  the  distance  through  which  P 
acts  is  twice  the  height  through  which  the  weight  rises, 
.\  s  =  U,     and      W  =  2P,     or    F  =:  IW, 


T 


(S\ 


i 


A 


w 

Fig.  168. 


Fio.  169. 


Fio.  170. 


3.  Single  Movalle  Pulley  with  Strings  not  Parallel. 
The  effective  component  of  the  power  is  P  cos  a  (Fig. 
171),  and  since  s  =  2h,  as  above, 

W=2P  cos  a, 

4.  First  System  of  Pulleys.  Suppose  that  the  weight 
(Fig.  172)  is  raised  through  a 
height  h,  the  wheel  D  is  evidently 
raised  2h;  the  wheel  C,  27^;  the 
wheel  B,  27^;  the  wheel  A,  2%; 

^■F  consequently  the  power  P  must 
act  through  a  distance  s  =  2%\ 
that  is,  W.li  =  2^A.P,  and  W  =: 
2^P;  or,  in  general, 


W=  2\P. 


Fio.  171. 


6.  Second  System  of  Pulleys. 
If  the  weight  is  raised  (Fig.  173)  a  height  h,  each  of 
the  6  strings  must  be  shortened  by  an  equal  amount; 
conseouently  the  distance  s  through  which  the  power 


PULLEY. 


233 


must  act  is  equal  to  6A.     Hence  P,Qh  =  W.h,   and  W 
=  6P;  or,  in  general, 

W=nP, 

6.  TJiird  System  of  Pulleys.  Suppose  that  tlie  weight 
is  raised  (Fig.  174)  a  height  h;  then,  since  the  pulley 
D  is  fixed,  the  pulley  C  will  on  this  account  move  down 
a  distance  h;  but,  since  the  point  c  also  rises  a  height  h, 


\s«s5a^r 


the  pulley  ^  will  move  through  a  distance  2(A)  +  h. 
Similarly,  the  motion  of  A  will  be  through  a  distance 
2{2h  +  7i)  +  7i,  and  the  distance  {=  s)  lor  P  = 
2(2'h  +  2A  +  70  +  A  =  15,  or  (2*  -  l)h.  Therefore 
Wh  =  (2*  -  l)h.P,  and  W=  (2*-l)P;  or,  in  gene- 
ral, 

TT  =  (2«  -  1)P. 


226.  Application  of  the  Pulley.     The  second  system 
of  pulleys  (222)  is  tliat  which  is  most  frequently  em- 


234 


STATICS. 


ployed 
wheels 


in  practice.     For  the  sake  of  compactness  the 
are  arranged   side  by  side  in  each  of  the  two 
blocks,  as  shown  in  Fig.  175.     The  relation 
of  P  to  W  is  the  same  as  in  Fig  163. 

The  theoretical  relation  of  P  to  TF  is  not 
practically  attained,  for  both  friction  and  the 
stiffness  of  the  rope  are  very  serious  resist- 
ances to  be  overcome.  For  many  purposes, 
notwithstanding  this  loss,  the  pulley  is  a 
most  useful  mechanical  contrivance.  It  is 
jp  often  used  in  connection  with  the  wheel  and 
axle  or  with  toothed  wheels,  as  in  derricks 
and  cranes.  It  plays  an  important  part  in 
the  rigging  of  a  ship.  The  single  fixed 
pulley  (218)  is  often  employed  where  it  is  de- 
sired to  change  the  direction  of  the  force; 
e.g.,  in  the  case  of  a  well. 

EXAMPLES. 
XXX.  PuUei/.    Articles  317-226. 

1.  A  man  weighing  150  lbs.  sits  on  a  platform 
suspended  from  a  movable  pulley  {B,  Fig.  IGO,  p.  228), 
and  raises  himself  by  a  rope  passing  over  a  fixed 
pulley  (A):  Supposing  the  strings  all  parallel,  (a) 
what  force  does  he  exert?  (b)  What  upward  force 
is  needed  if  the  rope  passes  under  a  pulley  fixed  to 
the  ground  before  coming  to  his  hand? 

2.  In  a  combination  of  pulleys,  as  in  Fig.  162,  W  = 
1152  lbs.  and  P=72  lbs. :  How  many  pulleys  are  there? 

3.  In  a  combination  of  pulleys,  as  in  Fig.  163,  TT  = 
336  lbs.  and  P  =  42  lbs.:  How  many  movable  pulleys  are  there? 

4.  In  a  combination  as  in  Fig.  164,  TT  =  840  lbs.,  P  =  56  lbs. : 
What  is  the  number  of  pulleys? 

5.  In  the  single  movable  pulley,  P  =  100  lbs. :   Calculate  the 
value  of  Tf  if  2a  =  30°,  =  60%  =  120%  =  150%  =  180% 


Fio.  175. 


227.] 


INCLINED  PLANE. 


235 


6.  What  force  is  needed  to  support  500  lbs.  by  the  first  system 
of  pulleys,  there  being  4  in  all?  What  is  the  force  if  each  pulley 
weighs  i  lb.  ? 

7.  Find  P  as  in  example  6,  if  the  second  system  of  pulleys  is 
employed. 

8.  Find  P  as  in  example  6,  if  the  third  system  is  used. 

9.  What  is  the  relation  of  Pto  TTin  Fig.  165,  if  the  weights  of 
the  movable  pulleys  are  taken  into  account? 

10.  What  is  the  relation  of  P  to  TF  in  Fig.  167,  if  the  weights  of 
the  movable  pulleys  are  considered? 

V.  Inclined  Plane. 

227.  The  Ii^clined  Plake,  considered  as  one  of  the 
simple  machines,  is  a  rigid  plane  inclined  to  the  horizon 
at  an  angle  a,  and  upon  it  a  weight  is  supported  by  a 
power  acting  in  some  definite  direction.  If  a  section 
be  made  perpendicular  to  the  plane,  the  figure  below 
(176)  is  obtained.   Here  HL  is  the  length  {I)  of  the  plane, 


HK  its  height  (h),  and  LK  its  base  {h).  The  three 
forces  acting  upon  a  body,  and,  as  we  suppose,  holding  it 
in  equilibrium,  are  the  weight  {W)  acting  vertically 
downward,  the  power  (P)  acting  at  some  angle  /?  with 
the  plane,  and  the  resistance  of  the  plane  acting  at  right 
angles  to  it;  these  forces  are  supposed  to  act  in  the  same 
plane. 


236 


STATICS. 


[228. 


228.  Relation  of  P,  "W,  and  R.  First  Method.  If  the 
three  forces  P,  W,  R,  acting  together  at  0,  are  in  equi- 
librium, then  (133)  each  force  is  proportional  to  the 
sine  of  the  angle  between  the  directions  of  the  other 
two.     That  is, 

P:  W:  R=^m  WOE  :  sin  POE  :  sin  WOP, 

=  sin  (180°  -  a)  :  sin  (90°  -  /?)  :  sin  (90°  +  a  +/?), 

=  sin  Of :  cos  fi  :  cos  (a  -\-  fi). 


Hence 


W  sin«       „ 


cos  /3 


Wcos(a-\-P) 
cos  /3 


Second  Method.  The  above  relation  may  also  be  ob- 
tained as  follows:  Since  the  forces  P,  W,  E  are  in  equi- 
librium (141),  the  algebraic  sum  of  their  components 
along  any  two  lines  at  right  angles  to  each  other  will  be 
equal  to  zero. 

Take  as  these  directions  (Fig.  177)  a  line  parallel  to 
the  length  of  the  plane,  and  one  perpendicular  to  it 


coinciding  with  the  direction  of  E.  Then,  taken  geo- 
metrically, the  component  of  E  along  HL  =  0,  of  P  = 
ab»  of  IF  =  —  ad',  also,  along  the  other  axis  the  compo- 


229.]  INCLINED   PLANE.  237 

nents  of  R,  P,  and  W  are  respectively  R,  ac,  —  ae.     Ex- 
pressing these  conditions  trigonometrically,  we  have,  first, 

P  cos  /?  —  W  mi  a  =  0, 

„        TT  sin  ^  ,^ . 

or  P  = VT-;  (1) 

cos  p  ^  ' 

and  second,     P  +  P  sin  /?  —  TT  cos  or  =  0, 

or  R  =:W  cos  a  —  P  sin  /?. 

Substituting  the  value  of  P  from  (1)  in  the  preceding 
equation,  we  obtain 

„        Txr  W  sin  r>  sin  6 

R  =  W  cos  a ^ — ^, 

cos  p 

Tr(cos  a  cos  y5  —  sin  o'  sin/?) 


cosy6? 
Tfcos(«r  +  /?) 

cos  p 


(3) 


229.  Special  Cases.  The  values  of  P  and  R  in  terms  of 
W  and  the  angles  a  and  /?,  derived  in  Art.  228,  apply  to 
all  cases,  whatever  the  direction  of  P.  If  now  the  power 
acts  along  the  plane,  or  horizontally,  these  general  equa- 
tions take  a  special  form  applicable  to  the  particular  case. 

(a)  The  power  acts  along  the  plane  (Fig.  178).  Here 
/5  =  0,  and  cos  /?  =  1 ;  hence,  from  the  general  value 

„        Wsm  a  ... 

P  = ^— ,  we  obtain 

cos  yo 

P  =  Wsina,  (3) 

and,  from  the  general  value  of 

Wcos{a^/3)  ,  ,  . 

R  = —'       ,  we  obtain 

cos /J 

R=Wcosa,  (4) 


238 


STATICS. 


[230. 


(b)  The  power  acts  liorizontdlly   (Fig.   179). 
P  =z  —  ay  Qo^  [—  a)  =  cos  a.     Hence,  for 

^        W  sm  a 


Here 


cosyS   ' 


is  obtained 


Tfsinar 


P  = =  Wisina, 


and,  for 


Ji 


Wcos  {CX+/3) 


cos  /3 


cos  a 


is  obtained 


(5) 


11  = 


W 


cos  a 


W  sec  a. 


(6) 


From  (5),  if  «  =  90°,  P  =  oo  ;  that  is,  no  finite  force 
can  support  a  body  against  a  vertical  surface  if  the  sur- 
faces in  contact  are  perfectly  smooth  and  there  is  no 
adhesion.  This  is  only  a  special  case  of  the  general 
principle  that  the  action  of  a  force  does  not  affect  the 
motion  of  a  body  in  a  direction  at  right  angles  to  that  in 
which  it  acts. 

230.  The  results  in   {a)   and   (b)   of  the  preceding 


article  can  also  be  obtained  independently  by  another 
method. 

(a)  The  power  acts  along  the  plane.      Let  the  lines 
P,  R,  W  represent  the  three  forces  holding  the  body  at 


230.J 


INCLINED  PLANE. 


239 


A  in  equilibrium.  From  P  draw  BC  parallel  to  the 
direction  of  IT;  then  the  triangle  ABC  has  its  three 
sides  respectively  parallel  to  the  three  forces,  and  hence 
(132,  Cor.)  these  sides  are  proportional  to  them.  Again, 
the  triangles  ABC  and  KHL  are  mutually  equiangular 
and  similar,  hence 

F:  W:R  =  AB'.BC:  AG, 


or 


and 


=  HK :  HL  :  LK\ 

p 

w  ~ 

HK 

R 

W 

LK 

(1) 


(2) 


The  result  in  (1)  is  sometimes  stated  in  this  form: 
When  the  Poiver  acts  along  the  plane,  the  Power  is  to  the 
Weight  as  the  height  of  the  plane  is  to  the  length, 

(b)  The  power  acts  horizontally.     Let  (Fig.  179)  the 


Fig.  179. 


three  forces  P,  R,  W  be  represented  by  the  three  lines 
meeting  at  A.  Through  B  draw  BC  parallel  to  the  di- 
rection of  W,  and  produce  R  to  meet  BC  at  C.  The 
sides  of  the  triangle  ABC  are  respectively  parallel  to  the 
three  forces  P,  R,  W,  and  therefore  (132,  Cor.)  are  pro- 


240 


STATICS. 


[231. 


portional  to  them;  moreover,  ABC  and  HKL  are  similar. 
Hence 

P  :  W :  B  =  AB  :  BC :  AC  =  HX :  KL  :  HL, 
P        HK 


or 


W 

A 

W 


KL 

HL 
KL 


=  tan  a, 


=  sec  or. 


(3) 


(4) 


The  result  in  (3)  may  be  stated  as  follows:  When  the 
Power  acts  horizontally,  the  Power  is  to  the  Weight  as  the 
height  of  the  plane  is  to  the  base, 

231.  Inclined  Plane  on  the  Principle  of  Work.  Let 
(Fig.  180)  the  power  P,  acting  at  an  angle  (/3)  with  the 


Fig.  180. 

plane,  raise  the  weight  ( W)  from  L  to  IT.  The  effective 
component  of  the  force,  parallel  to  the  plane,  is  P  cos  JS, 
and  the  distance  through  which  it  acts  is  the  length  of 
the  plane  (/);  therefore  the  work  done  by  the  power  is 
P  cos  /3,l,  that  done  upon  the  weight  is  Wh,     Hence 

P  cos  /3.1  =  Wh,        but        h  =  I  sin  a; 

,\  P  cos  /3  =W  sin  a. 


232.]  INCLINED  PLANE.  241 

(a)  The  power  ads  along  the  plane.  For  this  case  we 
have 

PJ  =  W.h,        or        P.l  =  WJ  sin  a, 

and  P=  W  sin  a. 

(Z>)  The  power  acts  horizontally.  The  effective  com- 
ponent of  P  is  P  cos  (  —  a)  =  Feoscx;  hence,  as  before, 

P  cos  a.l  =  JV.h  =  W.l  sin  a, 
P  cos  a  =  W  sin  a,         P  =  W  tan  a, 

232.  Applications  of  the  Inclined  Plane.  An  applica- 
tion of  the  inclined  plane  is  seen  in  the  arrangement  by 
which  boxes  or  barrels  are  pushed  up  from  the  ground 
into  a  wagon.  A  carriage-road  leading  gradually  up  a 
mountain,  or  a  railroad  on  an  up-grade,  are  other  ex- 
amples. If  the  power  acts  parallel  to  the  plane,  as  is 
generally  the  case,  it  has  to  support  only  one  component 
of  the  weight  ( W  sin  a)  at  any  one  time  (neglecting  fric- 
tion). It  is  evident,  however,  that  the  intervention  of 
the  inclined  plane  does  not  diminish  the  amount  of  work 
to  be  done,  as  that  is  the  same  for  a  given  vertical 
height,  whatever  the  angle  of  inclination. 

The  wedge  (233)  is  sometimes  considered  as  a  com* 
bination  of  two  inclined  jolanes,  base  to  base.  The  screw 
may  be  considered  as  an  inclined  plane  wound  around  a 
cylinder. 

EXAMPLES. 

XXXI.  Inclined  Plane.    Articles  227-233. 

[The  plane  is  supposed  to  be  perfectly  smooth.] 

X.  The  angle  of  the  plane  is  20°,  the  weight  is  120  lbs. :  What 
force  is  required  to  support  the  weight  (a)  acting  parallel  to  the 


242  STATICS.  [232. 

plane?  (b)  acting  horizontally?  (c)  acting  at  an  angle  of  30°  with 
the  plane? 

2.  What  is  the  reaction  of  the  plane  in  the  three  cases  in  ex- 
ample 1  ? 

3.  A  force  of  100  lbs.  acts  parallel  to  an  inclined  plane :  What 
weight  can  it  support  in  the  following  cases — the  angle  of  the 
plane  is  (a)  10°,  (b)  30°,  (c)  45°,  (d)  60°,  (e)  80°,  (/)  90°? 

4.  A  weight  of  100  lbs.  rests  on  an  inclined  plane :  What  force 
acting  parallel  to  the  plane  is  required  to  support  it,  if  the  angle 
of  the  plane  has  the  same  values  as  in  example  3? 

5.  A  force  of  100  lbs,  acts  horizontally  to  an  inclined  plane: 
What  weight  can  it  support  in  the  different  cases  given  in  ex- 
ample 3? 

6.  A  weight  of  100  lbs.  rests  on  an  inclined  plane :  What  force, 
acting  horizontally,  is  required  to  support  it  in  the  several  cases 
of  example  3? 

7.  A  railroad  has  a  grade  of  88  feet  to  the  mile :  What  force 
must  the  locomotive  exert  to  support  the  weight  of  the  whole 
train,  taking  that  at  25  tons? 

8.  The  length,  height,  and  base  of  a  plane  are  in  the  ratio  of 
5:3:4.  Into  what  two  parts  may  a  weight  of  56  lbs.  be  divided 
so  that  one  part  resting  on  the  plane  may  be  supported  by  the 
other  hanging  over  the  top  vertically  downward? 

9.  If  a  horse  can  raise  600  lbs.  vertically,  what  weight  can  he 
raise  on  a  railway  having  a  grade  of  3°  ? 

10.  The  grade  of  a  railway  is  44  feet  to  the  mile :  What  power 
(acting  parallel)  is  required  to  support  any  given  weight? 

11.  A  body  is  supported  on  an  inclined  plane  by  a  force  of  40 
lbs.  acting  parallel  to  the  plane;  but  if  the  force  acts  horizontally 
it  must  equal  50  lbs. :  Required  the  weight  of  the  body,  and  the 
inclination  of  the  plane. 

12.  Two  inclined  planes  of  lengths  40  and  60  feet  are  placed  so 
that  they  slope  in  opposite  directions,  and  their  equal  heights,  12 
feet  each,  coincide ;  a  weight  of  8  lbs.  is  supported  in  the  longer 
plane  by  a  string,  parallel  to  the  plane,  passing  over  a  pulley  at 
the  top,  and  attached  to  a  second  weight  on  the  shorter  plane : 
What  is  the  second  weight? 

13.  Weights  of  8  and  12  lbs.  are  supported  in  equilibrium  on 
two  inclined  planes,  so  placed  that  their  equal  heights  of  6  feet 


t.] 


WEDGE, 


243 


each  coincide;  they  are  attached  to  the  extremities  of  the  same 
string  passing,  parallel  to  the  planes,  over  a  pulley  at  the  top : 
What  are  the  lengths  of  the  planes,  the  angle  ©f  the  first  plane 
being  30°? 

VI.  Wedge. 

233.  The  Wedge  in  its  simplest  form  is  a  five-sided 
solid,  of  wMcli  two  adjacent  sides  meeting  in  the  edge 
are  rectangles,  the  two  opposite  ends  are  triangles,  and 
the  back  is  a  rectangle.  The  power  is  supposed  to  act 
in  a  direction  perpendicular  to  the  back,  and,  assuming 
the  surfaces  in  contact  to  be  perfectly  smooth,  the  resist- 
ances are  felt  in  the  same  plane  perpendicular  to  the 
sides. 

234.  Suppose  the  triangle  in  Fig.  181  to  represent  the 
section  made   by  a  perpendicular   plane  through  the 


wedge;  then,  if  the  three  forces  P,  Q,  R  hold  the  wedge 
in  equilibrium,  their  lines  of  direction  will,  if  produced, 
all  meet  at  some  point  0.  Also,  since  these  forces  are 
by  supposition  at  right  angles  to  the  sides  of  the  triangle 


244 


STATICS. 


[235. 


ABC,  these  sides  will  be  respectively  proportional  to 
them;  that  is, 

P  \Q\R=AB  '.AC'.BC. 

If  (as  in  Fig.  181)  the  section  is  an  isosceles  triangle, 
AC  =  BC,  and  Q  =  R',  hence 


R 


AB 
AC 


But  AB  =  2AD;  and  it  ACB  =  2a,  AD  -  AC  sin  a, 
or  AB  =  2 AC,  sin  a-,  that  is, 


P_ 
R 


2^C^sin  a 
AXJ 


=  2  sin  a. 


It  appears  from  this  equation  that  the  mechanical  ad- 
vantage increases  as  the  angle  of  the  wedge  diminishes. 

235.  Wedge  on  the  Principle  of  Work,  Let  ABO 
(Fig.  182)  be  an  isosceles  wedge,  and  let  the  power  act- 
ing against  the  resistances  Q  and  R 
(=  2R)  force  the  wedge  uniformly 
in  a  distance  equal  to  DC.  The 
work  done  by  P  is  P,DC.  The 
effective  distance  through  which 
the  resistance  has  been  overcome  ia 
D'E.     Therefore 

P.DC  =  2R.D'E, 

D'E=  D'C  ^ma-, 

.',  P.DC=  2R.DCama, 

Pro.  182.  or  P  =  2R  sin  a. 

236.  Application  of  the  Wedge.     In  practice  the  rela- 
tion established  for  the  wedge  has  little  value,  for  the 


236.]  WEDGE.  245 

resistance  due  to  friction  is  enormous.  The  wedge, 
•however,  is  an  important  instrument;  it  appears  in  many- 
cutting  tools,  such  as  the  knife,  chisel,  axe,  plane,  and 
so  on.  It  should  be  noticed  that  in  the  case  of  the  plane, 
for  example,  if  used  for  cutting  soft  wood,  the  angle  is 
small  and  the  edge  sharp;  for  harder  wood  the  angle  is 
larger.  The  tool  for  planing  iron  has  a  very  lai'ge 
angle — varying,  say,  from  60°  to  80°. 

When  the  wedge  is  employed  as  for  cleaving  wood,  the 
resistance  due  to  the  cohesion  and  friction  combined  is  so 
great  that,  instead  of  the  pressure  supposed  in  the  above 
article,  a  blow  from  a  heavy  body,  as  an  axe,  is  used  to 
drive  it  in.  In  this  the  principle  explained  in  Art.  105  is 
employed;  the  energy  of  a  heavy  body  in  motion  being 
expended  through  a  very  small  distance,  and  hence  over- 
coming a  great  resistance.  A  nail  is  a  familiar  form  of 
wedge,  and  its  use  further  illustrates  this  principle ;  it 
depends  upon  friction  for  its  hold  in  the  substance  into 
which  it  has  been  driven. 

EXAMPLES. 
XXXII.  Wedge.    Articles  233-235. 

1.  A  wedge  is  isosceles  in  shape  and  has  an  angle  of  20' ;  if 
P  =  40  lbs.,  what  is  the  resistance  on  each  face? 

2.  A  wedge  is  isosceles,  and  the  angle  90° ;  a  force  of  100  lbs. 
acts  at  the  back:  What  are  the  other  two  forces  ? 

3.  A  wedge  is  isosceles,  the  power  acting  on  the  back  is  40  lbs., 
and  the  forces  on  the  other  sides  are  60  lbs.  each:  What  is  the 
angle  of  the  wedge? 

4.  A  wedge  is  isosceles  and  has  an  angle  of  60° :  What  is  the 
relation  between  tlie  three  forces? 

5.  The  wedge  is  right-angled  and  the  three  sides  have  lengths 
of  15,  12,  and  9  (bcack):  If  P  =  100,  what  are  the  other  two  forces? 

6.  The  angle  of  tlie  wedge  is  30°,  the  back  is  10,  and  one  side  is 
20:  What  is  the  ratio  of  the  three  forces? 


246  STATICS.  [237. 


VII.  Screw. 

237.  The  Screw  consists  of  a  solid  cylinder  with  a 
raised  portion  passing  spirally  about  it,  which  is  called 
the  thread.  This  thread  is  either  rectangular  or  tri- 
angular in  shape.  It  may  be  regarded  as  generated  by 
the  revolution  of  a  rectangle,  in  the  one  case,  or  an 
isosceles  triangle,  in  the  other,  about  the  cylinder  at  the 
same  time  that  it  advances  uniformly  parallel  to  the 
axis,  and  at  such  a  rate  that  in  each  revolution  it  goes  a 
distance  equal  to  its  own  width.  The  two  kinds  of 
threads  are  shown  in  Figs.  183,  184. 

The  screw  in  use  works  in  a  nut  whose  parts  are  com- 


Fia.  183.  Fig.  184. 

plementary  to  those  of  the  screw,  so  that  the  one  fits 
closely  into  the  other.  The  power  acts  at  the  end  of  a 
lever-arm  to  turn  the  screw  in  the  nut;  either  one  may 
be  made  stationary,  so  that  the  other  moves  with  refer- 
ence to  it.  The  pressure  of  the  weight  or  resistance  is 
felt  in  the  direction  of  the  longer  axis  of  the  screw,  but 
the  resistance  between  the  screw  and  nut  is  felt  at  each 
point  of  contact  between  them  and  perpendicular  to 
their  common  surface. 

238.  In  the  screw  the  Power  is  to  the  Weight  as  the 
distance  between  the  threads  is  to  the  circumference 
described  by  the  Power, 

Suppose  the  surface  of  the  cylinder,  about  which  the 


S38.] 


SCEEW. 


247 


thread  passes  as  a  spiral  line  (its  thickness  being  neglect- 
ed), to  be  unrolled  on  a  plane.  A  rect- 
angle with  a  series  of  equal  triangles 
is  the  result,  as  shown  in  Fig.  185. 
Here  AB  is  the  circumference  of  the 
cylinder  =  27rr,  if  r  is  the  radius  of 
the  cylinder;  CAB  is  the  angle  between 
the  thread  and  a  horizontal  line,  called 
the  angle  of  the  screw;  BG  is  equal  to 
the  distance  between  the  threads,  also 
called  the  pitch  of  the  screw. 

BC  -  AB  tan  «; 

.  •.  distance  between  threads  =  2;rr  tan  a. 

Again,  at  each  point  of  contact  between  the  screw  and 
the  nut  a  force  acts  horizontally 
and  holds  both  (1)  P,  acting  on  the 
lever-arm  R  (Fig.  186),  and  (2)  W, 
acting  verfcically  downward  in  equi- 
librium. Let  the  sum  of  these 
partial  horizontal  forces  be  repre- 
sented by  i^;  then,  taking  moments 
about  the  axis, 


P.R  =  F.r, 


(1) 


•*r^ 


Fig.  186. 


For  P  and  F  tend  to  turn  the 
cylinder  in  opposite  directions. 
Also,  on  the  principle  of  the  in- 
clined plane,  since  the  force  F 
acting  horizontally  [229  (5)]  sup- 


ports the  total  weight  TF, 

F  =  TF.tan  or. 


(2) 


248  STATICS.  [23a 

From  (1)  and  (2) 

P.R  =  W.r  tan  a, 

P         r  tan  or 


Inultiplying  by  2;r, 


W  ~       R     ' 

P  _  27rr  tan  a 
W  ~       27rE    ' 


But  27tr  tan  a  =  distance  between  the  threads,  and 
27rB  is  the  circumference  described  by  the  power-arm; 
hence  the  relation  already  given  is  deduced. 

239.  Screw  on  the  Principle  of  Work.  Let  the  power 
acting  on  its  lever-arm  cause  it  to  describe  a  complete 
circumference  27tR;  the  work  done  by  the  power  is  then 
P.27tR,  At  the  same  time  the  weight  has  been  raised 
(or  resistance  overcome)  through  a  distance  equal  to 
that  between  two  consecutive  threads.     Therefore 

P.27tR  =  Tf  X  dist.  between  threads, 

P  _  dist.  between  threads 
W  ~  2^  • 

240.  Application  of  the  Screw.  The  screw  is  practi- 
cally a  most  important  mechanical  instrument,  being 
used  in  many  cases  where  a  great  weight  is  to  be  raised 
or  a  heavy  pressure  to  be  exerted.  For  example,  build- 
ings are  often  raised  by  the  combined  use  of  a  number 
of  screws,  and  screw-presses  are  employed  for  many  dif- 
ferent purposes. 

By  increasing  the  length  of  the  power-arm,  or  dimin- 
ishing the  distance  between  the  threads,  any  required 
mechanical  advantage  may  theoretically  be  obtained  ;  in 
fact,    however,  a  limit  is  soon  reached;  friction  is  a 


242.]  SCEEW.  249 

serious  element,  and  the  modulus  of  the  machine  is 
small.  The  use  of  the  common  screw,  as,  for  example, 
in  binding  two  boards  together,  depends  for  its  efficiency 
entirely  upon  friction. 

241.  Micrometer  Screw.  The  screw  is  also  employed 
for  measuring  very  small  distances,  and  is  then  called  a 
micrometer  screw.  In  this  case  the  relation  in  velocity 
of  motion  of  the  parts  is  the  matter  considered,  and  the 
relation  of  P  to  W  is  lost  sight  of.  The  principle  of  the 
micrometer  screw  will  be  clear  from  the  following 
remarks.  Suppose  a  screw  with  100  threads  to  the 
inch:  it  is  obvious  that  each  complete  revolution  will 
advance  the  screw  if  the  nut  is  stationary,  or  the  nut  if 
the  screw  is  held  firm,  through  a  distance  of  yj^  of  an 
inch.  Suppose,  further,  that  the  head  of  the  screw 
is  a  circle  whose  circumference  is  graduated  into  100 
equal  parts:  then,  if  it  is  arranged  with  a  fixed  index,  it 
is  easy  to  turn  the  head — that  is,  the  screw — through  j^ 
of  a  revolution,  and  this  will  cause  an  advance  of  yoo-  of 
Y^Tj-  of  an  inch;  that  is,  -^qIqq  of  an  inch  for  the  screw 
itself.  Screws  with  very  fine  threads,  and  hence  giving 
very  slow  motion,  find  many  applications  in  physical 
apparatus. 

242.  Differential  Screw.  The  differential  screw  gives 
a  greater  mechanical  advantage,  and  hence  a  slower 
motion  (which  may  be  the  end  desired),  than  can  be 
conveniently  obtained  from  the  simple  form.  Here  a 
larger  screw  turns  in  a  fixed  nut,  and  a  smaller  One  with 
a  less  pitch  turns  inside  of  it.  The  power  acts  directly 
on  the  larger  screw,  and  the  resistance  is  felt  by  the 
smaller.  Hence  it  is  evident  that  while  the  large  screw 
descends  the  small  screw  ascends,  and  the  actual  motion 


250  STATICS.  [243. 

of  the  platform  is  the  resultant  of  these  two  opposite 
velocities. 

The  relation  of  P  to  IT  is  given  immediately  by  the 
principle  of  work.  Suppose  the  power  to  act  through 
one  circumference  {27rE);  the  larger  screw  will  descend 
a  distance  equal  to  the  distance  between  its  threads  (p), 
and  the  smaller  screw  will  ascend  the  distance  between 
its  threads  (p') ;  hence  the  difference  of  these  two  will 
represent  the  distance  through  which  weight  is  raised 
(or  resistance  moved). 

W  ~   s  ~    ^TtR  ' 

The  Power  is  to  the  Weight  as  the  difference  of  the 
distances  hettveen  the  threads  of  the  two  screws  is  to  the 
circumference  described  hy  the  power-arm, 

243.  Endless  Screw.   Fig.  187  represents  what  is  called 


Fig.  187 


an  endless  screw.  Here  there  is  a  cylinder  with  a  uni- 
form thread  fitting  into  the  teeth  of  a  toothed  wheel. 
The  number  of  threads  in  the  screw  of  the  cylinder  AB 


248.]  SCEEW.  261 

will  determine  the  number  of  teeth  of  the  wheel  (7, 
which  will  be  advanced  in  one  revolution  of  the  former. 
Upon  this  depends  the  mechanical  advantage  of  the 
arrangement. 

EXAMPLES. 
XXXIII.  Screw.    Articles  236-342. 

1.  What  weight  can  be  raised  by  a  power  of  30  lbs.  acting  on  a 
iever-arm  2  feet  long,  if  the  screw  has  3  threads  to  the  inch? 

2.  If  a  power  of  40  lbs.  acting  on  an  arm  25  inches  long  can 
support  a  weight  of  8000  lbs. ,  what  will  be  the  distance  between 
the  threads  of  the  screw? 

3.  A  screw  has  10  threads  to  the  inch;  the  circumference 
described  by  the  power  is  4  feet :  What  power  is  needed  to  sup- 
port a  weight  of  6000  lbs? 

4.  While  the  point  of  application  of  the  power  makes  a  revo- 
lution of  3  feet,  the  screw  advances  i  of  an  inch;  the  power  is  50 
lbs. :  What  is  the  weight  raised? 

5.  The  angle  of  the  screw  is  10°,  and  the  length  of  the  power- 
arm  is  twenty  times  the  radius  of  the  cylinder:  What  is  the  me- 
chanical advantage? 

6.  The  circumference  described  by  the  power-arm  is  20  feet,  and 
the  mechanical  advantage  480 :  How  many  threads  in  the  screw 
are  there  to  the  inch  ? 

7.  The  circumference  described  by  the  power-arm  is  14  feet,  the 
power  is  60  lbs.,  and  the  weight  6  tons  (6  X  2240  lbs.):  What  is 
the  distance  between  the  threads  of  the  screw? 

8.  The  power-arm  of  a  differential  screw  is  18  inches;  there  are 
6  threads  to  the  inch  in  the  larger  screw,  and  8  threads  in  the 
smaller;  the  power  is  30  lbs. :  What  weight  can  be  supported? 


CHAPTER  IX.— PENDULUM. 


244.  Motion  in  a  Vertical  Circle.  Let  ^^(7  (Fig.  188) 
represent  a  yertical  circle,  regarded  as  perfectly  smooth. 
Suppose  a  particle  to  start  from  rest  at  A  and  slide  down 


toward  B ;  its  velocity  (v)  at  any  point  M  will  be  the 
same  (40)  as  if  it  had  fallen  through  the  vertical  height 
UF;  that  is, 

v'  =  2g.ER  (1) 

Since  DMB  and  DAB  are  right  angles,  by  geometry, 
MB"  =  DB.FBy  (2) 

AB'  =  DB.FB,  (3) 

Let  AB  =  a,  and  MB  =  z;  also,  DB  =  2r;  substitut- 
ing these  values  and  subtracting  (2)  from  (3), 

a'  -z'  =  2r  (EB  -  FB)  =  2r.BF.         (4) 


245.]  SIMPLE  PENDULUM.  253 

Introducing  the  value  of  EFimm  (4)  in  (1),  we  obtain 


t;'  =  I  {a'  -  z"),         V  =  ±  /|  («»  -  z'),    (5) 

For  the  point  A  (or  A'),  z  =  a,  and  therefore  v  =  0; 
for  B,  z  =  0  and  v  =  ±  o^  y  - ,  the  double  sign  indicat- 
ing that  the  motion  may  be  from  A  toward  ^'  (+),  or  the 
reverse  (— ).  For  two  points  M  and  M',  equally  dis- 
tant from  B,  BM  —  -\-  z,  and  BM'  =  —  z]  for  both 
these  the  value  of  v  is  the  same,  and  for  each  there  is  a 
-\-  value  and  a  —  value,  according  to  the  direction  of 
the  motion.     It  is  evident  that  if  the  particle  were  pro- 


jected from  B  with  a  velocity  equal  to  a  y  — ,  it  would 
ascend  to  A'  before  coming  to  rest. 

246.  Motion  of  a  Simple  Pendulum.  An  ideal  simple 
pendulum  is  a  material  particle  attached  by  a  string 
without  weight  to  the  point  of  suspension,  and  vibrating 
without  resistance  from  friction  or  any  other  source.  A 
particle  so  suspended  will,  if  set  in  motion,  continue  to 
vibrate  to  and  fro  in  an  arc  of  a  circle,  and  will  follow 
the  same  laws  as  the  body  descending  the  smooth  curve 
considered  in  the  previous  article.  The  tension  of  the 
string,  like  the  resistance  of  the  plane,  is  always  equal 
to  one  component  of  the  weight,  and  is  in  both  cases 
exerted  at  right  angles  to  the  direction  of  motion,  and 
hence  does  not  influence  the  velocity  of  the  particle. 

Therefore  the  value  of  the  velocity,  v=  y  -  {a^  —  z^)y 
obtained  in  Art.  244  will  apply  also  to  the  pendulum  at 


254 


PENDULUM. 


[245. 


any  point  in  its  course.  If  now  the  radius  r  (the  length 
of  the  pendulum)  is  great  and  the  length  of  the  arc  of 
vibration  is  very  small,  we  may  take  a  and  z  in  this 
value  of  V  as  representing  the  arcs  instead  of  the  chords, 
and  this  is  assumed  in  the  following  demonstration. 
The  error  arising  from  this  assumption  may  be  neglected 
without  destroying  the  value  of  the  result. 

Let  aaf  (Fig.  189)  represent  the  arc  of  vibration  of  the 
simple  pendulum,  whose  length  (r)  is  CB,  Take  the 
straight  line  AA'  equal  to  aa' , 
and  so  that  every  point  m  on  the 
arc  has  a  corresponding  point  M 
on  the  straight  line.  AYe  may 
without  error  imagine  the  pen- 
dulum to  vibrate  from  A  to  A^, 
following  the  same  law  as  in  the 
arc  aa\  so  that  its  velocity  at 
any  point  M  will  be  expressed  by 
the  above  formula, 


y^K-. 


when  AB  =  BA'  =  a,  and  BM 
=  z.  Here,  as  remarked  above, 
a  and  z  are  arcs,  not  chords. 

Upon  AA^  as  a  diameter  de- 
scribe the  circle  AJVA',  and  sup- 
pose a  particle  to  move  uniformly 
about  the  semi-circumference 
ANA'  with  the  constant  velocity 

ay  - ,      Then,  since  the  dis- 


riG.  189. 


tance  passed  over  is  na,  the  time 
(0  required  for  this  imaginary  particle  to  go  from  A  to 


245.]  SIMPLE  PENDULUM.  255 

A',  since  the  distance  and  velocity  are  known,  is  (19)  as 
follows  : 

.  _  s  _       Tta     _        jJr_ 


It  will  now  be  shown  that  this  expression  also  gives  the 
time  required  by  the  pendulum  to  vibrate  from  A  to  A' 
(that  is,  from  a  to  a'). 

Tlie  constant  velocity  V  of  the  imaginary  particle,  viz. 

a  y~,  may  be  represented  (20)  at  any  point  in  its  path, 

as  JV,  by  a  tangent  to  the  curve  JSfQ  l=aY-].    The 

horizontal  component  of  this  velocity  is  then  represented 
by  ^F. 

NP  =  J^Q.cos  QNP, 

=  a  |/^.cos  BJ^M, 
r 

But  a  cos  BNM  =  NM,  and 


NM=  ^NB  "  -  BM"  =   Va'  -  z\ 
Therefore  NP  =  f^  (a'  -  z'). 

Now,  this  expression  for  the  horizontal  component  of 
the  velocity  of  the  imaginary  particle  at  JSF  is  also 
the  value  of  the  velocity  of  the  pendulum  at  the  cor- 
responding point  M.  The  meaning  of  this  result  is 
as  follows:  If,  at  the  same  instant  that  the  pendulum 


256  PEl^DULUM.  [245- 

commences  to  vibrate  from  A  to  A',  the  imaginary 
particle  starts  from  A  about  the  semi-circumference 
with  its  constant  velocity,  the  velocity  of  the  pen- 
dulum at  any  point  (as  if)  will  be  the  same  as  the 
horizontal  component  of  the  particle  at  the  correspond- 
ing point  (as  N),  and  therefore  the  two  will  reach  A' 
at  the  same  time.  The  time  required  for  a  single  vibra- 
tion of  the  pendulum  is,  therefore,  the  same  as  that  for 
the  imaginary  particle  to  g*o  in  its  path  from  A  to  A';  but 


-1^, 


the  latter  has  been  shown  to  be  equal  to  tt  y  —,  hence 

this  expression  also  gives  the  time  of  the  pendulum.     If 
for  r  we  write  I,  the  length  of  the  pendulum,  we  obtain 


/~l 


g 

It  is  to  be  noted  that: 

(1)  The  time  of  a  vibration  is  independent  of  the 
length  of  the  arc,  when  it  is  taken  very  small.  That  is, 
the  pendulum  will  vibrate  through  an  arc  of  1,  2,  or 
3  degrees  in  sensibly  the  same  time.  Further,  when  the 
arc  of  vibration  is  constant  (as  in  the  clock)  the  times 
of  vibration  are  necessarily  the  same. 

The  motion  of  the  pendulum,  to  and  fro  in  a  small 
arc,  is  the  type  of  a  great  variety  of  isochronous  vibra- 
tions; for  example,  those  of  a  tuning  fork,  of  a  musical 
string,  of  the  particles  of  air  in  sound-waves,  and,  too,  of 
the  ether  (109)  particles  in  the  case  of  heat  and  light. 

(2)  If  g  is  constant — that  is,  for  the  same  point  on  the 
earth — tJie  time  of  vihration  varies  directly  as  the  square 
root  of  the  length  (t  a  VT,  ort^ocl).  If  a  given  pendulum 
vibrates  in  1  second,  one  four  times  as  long  will  vibrate 
in  2  seconds,  and  another  one  quarter  as  long  will 
vibrate  i  a  second. 


246.]  COMPOUND  PENDULUM.  251 

(3)  If  the  length  is  constant,  tJie  time  of  vibration 
varies  inversely/  as  the  square  root  of  the  force  of  grav- 

(4)  If  the  time  is  constant — that  is,  for  two  pendu- 
lums at  different  stations,  each  vibrating  in  1  second — 
the  length  varies  directly  as  the  force  of  gravity  {I  oc  g). 

The  relations  in  (3)  and  (4)  give  a  ready  method  of 
comparing  the  values  of  g  for  different  latitudes  or  for 
different  heights  above  the  sea-level,  as  further  explained 
in  Art.  248. 

246.  Compound  Pendulum.  The  simple  pendulum, 
used  in  the  preceding  demonstration,  is  an  ideal  form 
not  obtainable  for  actual  experiment.  Any  pendulum 
that  can  be  constructed  is  a  compound  pendulum,  con- 
sisting of  an  indefinite  number  of  material  particles 
rigidly  joined  together  and  vibrating  from  a  fixed  axis. 
In  order  to  apply  the  result  in  Art.  245  to  a  compound 
pendulum,  it  is  necessary  to  obtain  the  value  of  Z,  and 
for  this  end  use  is  made  of  the  principle: 

The  length  of  a  compound  pendulum  is  equal  to  the 
length  of  a  simple  pendulum  which  would  vibrate  in  the 
same  time. 

Consider  the  compound  pendulum  as  a  simple  metal 
rod:  it  is  obvious  that  the  particles  near  the  axis  of 
support,  called  the  axis  of  suspension,  tend  to  vibrate 
more  quickly,  and  those  farthest  from  it  more  slowly, 
than  they  can  do,  since  they  must  all  vibrate  in  the 
same  time.  It  is  obvious  that  there  must  exist  a  line  in 
the  rod  which  is  so  situated  that  the  motion  of  the 
particles  on  it  is  neither  quickened  nor  retarded  by  the 
rest ;  that  is,  these  particles  would,  if  alone,  vibrate  in 
the  same  time  as  that  of  the  bar  as  a  whole.     This  line 


258  PEl^DULUM.  [247. 

through  these  particles  and  pai'allel  to  the  axis  of  sus- 
pension is  called  the  axis  of  oscillation.  The  distance 
between  these  two  axes  is  the  length  of  the  compound 
pendulum  as  above  defined. 

247.  To  find  the  axis  of  oscillation,  use  is  made  of 
the  principle:  The  axis  of  suspension  and  axis  of  oscilla- 
tion are  inter changealle.    Therefore,  if  the  pendulum  be 
swung  on  one  axis  and  the  time  of  vibration  be  deter- 
mined, and  then  the  second  axis  be  found,  so  that 
|]      if  this  is  made  the  axis  of  suspension  the  time  of 
vibration  will  be  exactly  the  same,  the  latter  axis 
is  the  axis  of  oscillation.     The  truth  of  this  prin- 
ciple is  established  in  works  on  higher  Mechanics. 
The  peiyiulum  exhibited  in  Fig.  190  is  a  form 
devised  by  Kater :  a  and  h  are  the  two  axes  of 
suspension  and  oscillation ;   v  and  to  are  two 
slides,  the  position   of  which  may  be  adjusted 
until  the  condition  in  regard  to  the  equal  times  of 
vibration  for  the  two  axes  is  satisfied. 

248.  Application  of  the  Pendulum.  The  most 
important  application  of  the  pendulum  is  as  an 
instrument  for  determining  the  value  of  the 
acceleration  of  gravity  {g).  The  direct  determi- 
nation of  g  requires  the  observation  of  either 
the  velocity  acquired  (32  feet  per  second  in  a 
second)  or  the  space  passed  over  (16  feet  in  the 
first  second  from  rest)  by  a  body  (27)  falling  in  a 
vacuum.  This  method  is  obviously  impractica- 
Fiaiso.  ^^^^  j^  Attwood's  machine  (74)  be  employed, 
the  force  of  gravity  may,  as  it  were,  be  weakened  so  that 
the  velocity  acquired  and  space  passed  through  are  much 
less  than  for  a  body  falling  freely.  In  this  way  approxi- 
mate values  of  g  may  be  obtained. 


260.]  APPLICATION   OF  THE  PENDULUM.  259 

For  accuracy,  however,  the  seconds  pendulum  gives 
the  simplest  and  most  satisfactory  method  of  determin- 
ing the  value  of  g.     From  Art.  245  we  see  that 

t=7ty-,        or        9  = -f' 

For  the  seconds  pendulum,  g  =  nH, 

In  the  actual  determination  of  the  value  of  I  for  a 
pendulum  vibrating  in  seconds  on  any  point  of  the 
earth's  surface,  many  refinements  of  observation  are 
required.  It  is  sufficient,  however,  to  say  here  that 
when  the  experiments  are  carried  on  with  all  possible 
care,  and  when  the  many  necessary  corrections  have  been 
introduced,  a  very  high  degree  of  accuracy  is  obtainable 
for  the  value  of  Z,  and  consequently  of  g. 

249.  The  following  table  gives  the  lengths  of  the 
seconds  pendulum  and  the  value  of  g  (for  the  sea-level) 
for  some  important  points  on  the  earth.  Those  for  the 
equator  and  pole  are  calculated  from  an  equation  the 
constants  of  which  have  been  deduced  from  numerous 
pendulum  experiments  in  different  localities;  the  others 
are  from  direct  observations  (U.  S.  Coast  Survey) : 

Latitude.       in«J.fec.      y«taeoJJ. 
ond  per  second.       ^"  mcnes. 

Equator 0°  33.091  39.017 

New  York  (Hoboken) 40°  43'  32.161  39.103 

Paris 48°  50'  32.185  39.133 

London  (Kew) 51°  29'  32.193  39.142 

Berlin 52°  30'  32.195  39.144 

Pole 90°    0'  32.255  39.217 

250.  The  lengths  of  the  seconds  pendulum,  as  deter- 
mined by  numerous  experiments,  are  of  great  value,  as 
giving  the  means  of  comparing  the  intensity  of  the  force 


260  PENDULUM.  [261. 

of  gravity  at  the  different  stations.  By  this  means  the 
ellipticity  of  the  earth  has  been  determined.  A  formula, 
alhided  to  in  the  previous  article,  has  also  been  obtained 
which  gives  the  values  of  I  and  g  for  any  required  lati- 
tude. The  ellipticity  of  the  earth,  obtained  in  this 
way,  differs  somewhat  from  the  similar  result  from  the 
trigonometrical  measurements  of  long  arcs  of  meridians. 
Moreover,  the  values  of  I  and  g  derived  from  the  formula 
do  not  always  agree  with  those  obtained  by  direct  ex- 
periment as  closely  as  might  be  expected.  The  ex- 
planation of  the  variations  is  to  be  found  in  the  facts 
(1)  that  the  earth  is  rather  an  ellipsoid  than  a  spheroid, 
and,  further  (2),  that  the  density  of  the  earth's  crust  at 
different  points  is  not  uniform.  This  latter  point  has 
been  extensively  investigated  by  means  of  these  pendu- 
lum observations;  it  is  found, for  example,  that  the  attrac- 
tion on  the  coast  is  greater  than  in  the  interior,  and 
that  it  is  still  greater  on  islands  in  the  soa.  From  this 
it  is  argued  that  the  density  of  the  earth's  crust  under 
the  ocean  is  greater  than  the  average  of  that  forming 
the  dry  land.  Other  similar  results  have  also  been 
obtained  by  this  means. 

251.  It  has  also  been  proposed  to  make  the  pendulum 
a  basis  of  a  system  of  weights  and  measures,  on  the 
ground  that  it  would,  at  a  given  place,  be  a  standard 
which  could  be  at  any  time  replaced  if  others  were 
destroyed.  It  was,  for  example,  enacted  by  Parliament 
in  1824  that  the  length  of  the  standard  yard  should  bear 
to  the  length  of  the  seconds  pendulum  in  London  (in 
vacuum  and  at  the  sea-level)  the  ratio  of  36  :  39.1393. 
The  difficulty  in  obtaining  the  length  of  the  pendulum 
with  the  degree  of  accuracy  now  needed  for  a  standard 
of  measures  makes  the  relation  of  little  practical  valu© 


261.]  EXAMPLES.  261 

The  pendulum  finds  a  further  application  as  a  regu- 
lator for  clocks. 

EXAMPLES. 

XXXIV.  Pendulum.    Articles  244-247. 

[The  length  of  the  seconds  pendulum  at  New  York  is  about  39.10 
inches,  and^f  =  32.16.] 

1.  Kequired  the  length  of  a  pendulum  at  New  York  which 
will  vibrate  {a)  in  |  second,  (6)  in  2  seconds,  (c)  in  2|  seconds. 

2.  Required  the  length  of  the  seconds  pendulum   where  the 
acceleration  of  gravity  is  32.25  (that  is,  near  the  pole). 

3.  Required  the  length  of  a  pendulum  to  vibrate  in  2  seconds, 
where  the  value  of  g  is  32.1. 

4.  What  would  be  the  length  of  a  pendulum  to  vibrate  in 
1  second  at  the  surface  of  the  sun  (acceleration  of  gravity  =  28  5')? 

5.  How  many  beats  in  a  minute  would  a  pendulum  8  feet  long 
make  in  New  York? 

6.  If  a  pendulum  24  inches  long  vibrates  in  |  second,  what  is 
the  length  of  a  seconds  pendulum? 


ADDITIOl^AL    EXAMPLES, 

INTRODUCING  THE  METRIC  UNITS. 


I.  Uniform  Motion  of  Translation  or  Rotation.    Articles  17-21; 

pp.  9-12. 

1.  A  body  travels  10  meters  per  second:  How  far  will  it  go  in 
a  day  of  24  hours? 

2.  A  velocity  of  50  kilometers  per  hour  corresponds  to  a  rate 
of  how  many  meters  per  second? 

3.  A  man  walks  uniformly  6  kilometers  per  hour:  {a)  How 
many  decimeters  does  he  go  in  a  second?  (b)  How  many  meters 
in  a  minute? 

4.  Two  bodies  start  from  the  same  point  in  opposite  directions; 
the  one  moves  at  a  rate  of  3  meters  per  second,  the  other  at  a 
rate  of  30  kilometers  per  hour:  {a)  What  will  be  the  distance 
between  them  at  the  end  of  8  minutes?  (p)  When  will  they  be 
6  kilometers  apart? 

5.  How  far  will  the  bodies  in  the  preceding  example  be  apart 
at  the  end  of  the  same  time,  if  they  move  in  the  same  direction? 

6.  (a)  What  is  the  angular  velocity  of  a  revolving  wheel  having 
a  radius  of  1  meter,  if  it  makes  14  revolutions  per  second  (take 
Tt  =  Z\)1  (b)  How  far  (in  kilometers)  will  a  point  on  the  circum- 
ference travel  in  10  hours? 

II.  Uniformly  Accelerated  Motion    Articles  22-28 ;    pp.    13-20. 
A.  Falling  Bodies  (g  =  about  9.80  meters  at  New  York). 

[The  body  is  supposed  to  start  from  rest.] 
1.  Calculate  the  distances  fallen  through  (in  meters)  and  the 
acquired  velocities  in  1,  2,  3,  4  seconds  from  rest. 


264 

2.  A  body  falls  10  seconds:  Required  (a)  the  velocity  acquired; 
(b)  the  whole  distance  fallen  through;  (c)  the  space  passed  over  in 
the  last  second  of  its  fall. 

3.  A  body  has  fallen  through  90  meters :  Required  (a)  the  time 
of  falling;  (b)  the  final  velocity. 

4.  A  body  has  acquired  in  falling  a  velocity  of  73.5  meters  per 
second:  Required  {a)  the  time  of  falling;  (b)  the  distance  fallen 
through. 

5.  A  body  in  falling  passed  over  44, 1  meters  in  the  last  second : 
Required  (a)  the  time  of  falling;  (5)  the  distance  fallen. 

B.  General  Case. — Acceleration  —f. 

1.  A  body  moves  100  meters  in  the  first  5  seconds  from  rest : 
What  is  the  acceleration? 

3.  A  body  moves  10  meters  in  the  first  second:  {a)  What  is  the 
acceleration?  (6)  How  far  will  it  go  in  6  seconds?  (c)  What  will 
be  its  final  velocity  at  the  end  of  this  time? 

3.  The  acceleration  is  12  meters-per-second  per  second:  {a) 
What  velocity  does  a  body  acquire  in  5  seconds?  (p)  What  space 
does  it  pass  over? 

4.  A  body  moves  54  meters  in  3  seconds,  and  96  in  the  next  2: 
Is  its  motion  uniformly  accelerated? 

5.  A  body  passes  over  64  meters  in  4  seconds:  What  distance 
must  it  go  in  the  next  5  to  satisfy  the  condition  of  uniformly 
accelerated  motion? 

6.  The  velocity  of  a  body  is  increased  from  20  to  40  meters  per 
second  while  it  passes  over  30  meters :  What  is  the  acceleration  ? 

III.  Composition  of  Constant  Velocities.     Articles  29-37;  pp.  22-27. 

1.  The  velocity  of  a  steamboat  is  10  kilometers  per  hour,  that 
of  the  stream  is  8,  and  a  man  walks  the  deck  from  stern  to 
bow  at  the  rate  of  6 :  Required  the  actual  velocity  of  the  boat 
{a)  if  headed  up  stream,  and  ip)  down  stream;  also  {c,  d),  that  of 
the  man  in  each  case. 

2.  The  velocities  of  the  boat  and  stream  are  respectively  100 
meters  and  80  meters  per  minute,  and  the  boat  is  headed  directly 
across  the  stream  (Fig.  11):  (a)  What  will  be  the  actual  direction 
of  the  boat's  motion?  {b)  What  the  rate  of  its  motion?  (c)  How 
long  will  the  passage  take  if  the  stream  is  2  kilometers  in  width? 


INTRODUCING  THE  METRIC   UNITS.  265 

3.  The  velocities  of  boat  and  stream  are  as  in  2,  but  it  is  re- 
quired that  the  boat  shall  go  directly  across  from  A  to  C  (Fig. 
12):  (a)  In  what  direction  must  the  boat  be  headed?  (b)  What 
will  be  its  actual  velocity  across?  (c)  What  time  will  the  passage 
take,  the  width  being  as  in  2? 

4.  A  ball  on  a  horizontal  surface  tends  to  move  north  with  a 
velocity  of  12  meters  per  second,  and  east  with  a  velocity  of 
5  meters  per  second:  (a)  What  will  be  the  actual  velocitj^  and  (b) 
in  what  direction? 

5.  A  ball,  moving  north  at  a  rate  of  8  meters  per  second,  re- 
ceives an  impulse  tending  to  make  it  move  due  south-east  with 
the  same  velocity :  (a)  What  path  will  it  take,  and  (b)  at  what 
rate  will  it  move? 

6.  A  man,  skating  uniformly  at  a  rate  of  4  meters  per  second, 
projects  a  ball  on  the  ice  in  a  direction  at  right  angles  to  his 
motion  at  a  rate  of  3  meters  per  second:  What  is  (a)  the  actual 
rate,  and  (6)  the  direction  of  its  motion  (friction  neglected)? 

IV.  Besolution  of  Constant  Velocities.     Article  38;  pp.  28,  29. 

1.  A  ball  tends  to  move  in  a  certain  direction  at  a  rate  of  6 
meters  per  second,  but  it  is  constrained  to  move  at  an  angle  of 
60°  with  this  direction :  Required  its  velocity  in  the  latter  direc- 
tion. 

2.  A  ball  rolls  at  the  rate  of  6  meters  per  second  across  the 
diagonal  of  a  rectangular  room  ABCD  whose  dimensions  are 
15  X  20  (=  AB  X  AG):  What  is  its  rate  of  motion  parallel  to 
each  side? 

3.  A  bod}'-  moves  N.  30°  E.  at  a  rate  of  5  kilometers  per  hour : 
Required  its  rate  of  motion  northerly  and  easterly. 

4.  A  boat,  though  headed  directly  across  a  stream,  actually 
moves  diagonally  across  the  stream  at  an  angle  of  30°  (BAG,  Fig. 
11)  and  at  a  rate  of  15  kilometers  per  hour:  Required  (a)  the  rate 
of  the  boat,  and  (b)  of  the  current  taken  independently. 

V.  Falling  down  an  Inclined  Plane.     Article  40;  pp.  32,  33. 

[The  plane  is  supposed  to  be  perfectly  smooth,  so  that  there  is  no 
friction.] 

1.  The  angle  of  the  plane  is  30° :  Required  (a)  the  acceleratiou 


266  ADDITIONAL  EXAMPLES, 

down  the  plane;  (b)  the  distance  fallen  through  in  4  seconds;  (c) 
the  velocity  acquired. 

2.  The  height  of  the  plane  is  19.6  meters  and  the  length  78.4; 

(a)  What  is  the  time  required  to  reach  the  bottom?    (b)  What  is 
the  velocity  acquired? 

3.  The  angle  of  the  plane  is  45° :  Required  the  time  of  falling 
490  meters. 

4.  The  length  of  a  plane  is  630  meters;  a  body  falls  down  it  in 
30  seconds:  (a)  What  is  the  acceleration?  (b)  What  is  the  height 
of  the  plane? 

VI.  Bodies  projected  vertically  downward.    Articles  41,  42; 
pp.  34,  35. 

1.  A  body  is  thrown  vertically  down  with  an  initial  velocity  of 
12  meters  per  second :  Required  (a)  the  velocity  at  the  end  of  7 
seconds;  (b)  the  distance  fallen  through. 

2.  A  body  is  projected  down  with  an  initial  velocity  of  19.1 
meters  per  second:  (a)  How  long  will  it  require  to  fall  218 meters? 

(b)  What  velocity  will  it  then  have? 

3.  What  velocity  of  projection  must  a  stone  have  to  reach  the 
bottom  of  a  cliff  100  meters  high  in  3  seconds? 

4.  A  stone  is  dropped  from  a  bucket  which  is  descending  a 
shaft  at  the  uniform  rate  of  3.5  meters  per  second,  and  at  the 
moment  when  the  bucket  is  75  meters  from  the  bottom :  (a)  How 
far  will  they  be  apart  in  two  seconds?  (b)  When  will  the  stone 
reach  the  bottom? 

VII.  Bodies  projected  vertically  upward.    Article  44;  pp.  37-39. 

1.  The  velocity  of  the  projection  upward  is  49  meters  per 
second:  Required  (a)  the  time  of  ascent;  (b)  of  descent;  (c)  the 
height  of  ascent;  (d)  the  distance  gone  in  the  first  and  last 
seconds  of  ascent. 

2.  A  body  is  projected  up  with  a  velocity  of  42  meters  per 
second:  (a)  When  will  it  be  37.1  meters  above  the  starting-point? 

3.  What  velocity  of  projection  must  a  ball  have  in  order  to 
ascend  just  160  meters? 

4.  What  time  does  a  body  require  to  ascend  250  meters,  that 
being  the  highest  point  reached? 


INTRODUCING  THE  METRIC  UNITS.  267 

5.  A  ball  thrown  up  passes  a  staging  36.4  meters  from  the 
ground  at  the  end  of  2  seconds:  {a)  What  was  the  velocity  of  the 
projection?    (b)  When  will  it  pass  it  again? 

VIII.  Projected  up  or  down  a  smooth  Inclined  Plane.    Article  45; 

p.  39. 

1.  The  height  of  the  plane  is  105  meters,  the  length  is  420 
meters,  the  velocity  of  projection  down  is  20.3  meters  per  second: 
{a)  How  long  will  it  require  to  descend?  (6)  What  will  be  the 
final  velocity? 

2.  The  angle  of  the  plane  is  30°,  the  velocity  of  projection 
down  is  10  meters  per  second:  Required  {a)  the  velocity  at  the 
end  of  4  seconds;  (b)  the  distance  gone  through. 

3.  The  height  and  length  of  the  plane  are  160  and  320  meters 
respectively :  {a)  What  velocity  is  required  that  the  body  should 
just  reach  the  top?    (J)  What  time  is  needed? 

IX.  Bodies  projected  against  Friction.   Articles,  41,  42;  pp.  34,  35. 

[The  retardation  (or  minus  acceleration)  due  to  friction  takes  the 
place  of  the  /in  the  formulas  of  articles  42  and  44.] 

1.  A  body  projected  on  a  rough  horizontal  plane  has  at  starting 
a  velocity  of  40  meters  per  second,  but  loses  this  at  the  rate  of  4 
meters  for  each  succeeding  second :  {a)  What  is  the  retardation 
(minus  acceleration)  due  to  friction?  ib)  When  will  the  body 
stop?    (c)  How  far  will  it  have  gone? 

2.  The  retardation  due  to  friction  is  for  each  second  6  deci- 
meters per  second  for  a  given  sliding  body,  the  initial  velocity 
is  12  meters  per  second:  Required  («)the  time  it  will  continue  to 
slide ;  (6)  the  distance  it  will  go ;  (c)  its  velocity  at  the  end  of  3 
seconds. 

3.  If  the  retardation  of  friction  is  2  decimeters-per-second  per 
second:  {a)  What  initial  velocity  (in  kilometers  per  hour)  must  a 
body  have  in  order  to  slide  just  160  meters?  (6)  If  the  velocity  is 
doubled,  how  much  farther  will  it  go? 

X.  Projectiles.     Articles  47-51 ;  pp.  43-50. 

1.  The  initial  velocity  of  a  projectile  is  245  meters  per  second, 
and  the  angle  of  elevation  is  30° :  Required  (a)  the  time  of  flight; 
(6)  the  range. 


268 

2.  The  initial  velocity  is  140  meters  per  second :  What  angle  of 
elevation  will  give  a  range  of  1.5  kilometers?  Show  that  there 
are  two  answers. 

3.  The  angle  of  elevation  is  15° :  What  initial  velocity  is  re- 
quired that  the  range  should  be  4  kilometers? 

4.  A  rifle-ball  is  shot  horizontally  from  the  top  of  a  tower  44.1 
meters  high,  and  with  an  initial  velocity  of  400  meters  per 
second :  When  and  how  far  from  the  base  of  the  tower  will  it 
strike  the  horizontal  plane  below? 

5.  A  ball  is  thrown  horizontally  from  the  top  of  a  cliff  above 
the  sea;  it  strikes  the  water  in  5  seconds,  and  at  a  horizontal  dis- 
tance of  a  kilometer:  What  was  (a)  the  initial  velocity,  and  (b) 
what  was  the  height  of  the  cliff? 

XI.  Mass — Density —  Volume.     Article  56 ;  pp.  53,  54. 

1.  What  is  the  ratio  in  volume  of  a  piece  of  silver  weighing 
20  kilograms  and  having  a  density  of  10.5  (referred  to  water  as 
unity),  and  a  piece  of  iron  weighing  5  kilograms  and  having 
a  density  of  7  ? 

2.  What  is  the  ratio  in  weight  (that  is,  in  mass)  of  two  blocks 
of  stone,  one  having  a  volume  of  1  cubic  decimeter  and  a  density 
of  3,  the  other  a  volume  of  400  cubic  centimeters  and  a  density 
of  2.75  ? 

3.  If  a  liter  of  dry  air  at  0°  weighs  1.2932  grams,  and  a  liter  of 
water  at  the  same  temperature  weighs  999.88  grams,  what  is  the 
density  of  the  water  referred  to  that  of  air  as  unity  ? 

4.  What  is  the  weight  of  a  liter  of  mercury  at  100°,  the  expan- 
sion of  volume  from  0°  to  100°  being  in  the  ratio  of  1  :  1.0154  ? 
The  density  of  mercury  at  0°  is  13.6,  referred  to  water  as  unity. 

XII.  Force  of  Gravity.    Articles  63-65;  pp.  58-62. 

1.  At  what  distance  from  the  centre  of  the  earth  would  a  mass 
of  matter  weighing  16  kilograms  on  the  earth's  surface  exert  a 
full  equivalent  to  1  kilogram  on  a  spring-balance  ? 

2.  If  the  mass  of  the  sun  is  355,000  times  that  of  the  earth,  and 
its  diameter  112  times,  what  would  be  the  acceleration  of  gravity 
at  its  surface  ? 

3   If  the  moon's  mass  is  ^^^  of  that  of  the  earth,  and  its  dia- 


INTEODUCIl^^G  THE  METRIC   UNITS.  269 

meter  3476  kilometers,  that  of  the  earth  heing  about  12,715  kilo- 
meters, what  is  the  acceleration  of  gravity  on  the  moon's  surface? 
4.  If  the  distance  from  the  earth  to  the  moon  is  60  times  the 
earth's  radius,  what  is  the  force  of  the  earth's  attraction  at  the 
moon  ? 

XIII.  Collision  of  Inelastic  Bodies.     Article  70;  pp.  70,  71. 

[The  bodies  are  supposed  to  be  perfectly  inelastic,  their  motion 
is  uniform,  and  the  impact  is  direct.] 

1.  A  ball  weighing  10  kilos  and  having  a  velocity  of  6  meters 
per  second  overtakes  a  second  ball  weighing  5  kilos  and  whose 
velocity  is  3  meters  per  second:  What  is  the  final  velocity? 

2.  If  the  first  ball  in  the  preceding  example  meets  the  second, 
what  is  the  final  velocity? 

3.  A  body  weighing  40  kilos  strikes  another  at  rest  weighing 
360  kilos,  and  the  two  move  on  with  a  velocity  of  1  meter  per 
second:  What  was  the  original  velocity  of  the  first  ball? 

4.  Three  bodies,  each  weighing  4  kilos,  are  situated  in  a 
straight  line;  a  fourth,  weighing  8  kilos  and  moving  at  a  rate  of 
6  meters  per  second,  strikes  them  in  succession:  What  velocity 
results  after  each  impact? 

5.  A  rifle-bullet  weighing  30  grams  is  fired  into  a  suspended 
block  weighing  15  kilos;  the  blow  causes  the  wood  to  rise  36 
millimeters :  Required  the  velocity  of  the  bullet  at  the  moment 
of  impact. 

XIV.  Oeneral  Dynamical  Prdhlems.     Articles  68,  73-76";  pp.  66, 
67,  and  75-80. 

1.  If  in  Attwood's  machine  P=  102  grams  and  Q  =  45  grams: 
{a)  What  is  the  acceleration  ?  (6)  What  space  will  be  passed 
through  in  2  seconds  ? 

2.  («)  At  what  height  above  the  earth's  surface  woutd  a  body 
fall  625  millimeters  in  the  first  second  from  rest?  (&)  If  its  weight 
was  16  kilos,  what  pull  would  it  exert  on  a  spring-balance  at  this 
point? 

3.  A  weight  of  6  kilos  hanging  over  the  edge  of  a  smooth  table 
drags  a  weight  of  15  kilos  with  it :  What  is  the  acceleration  and 
the  tension  of  the  string? 


270  ADDITIONAL  EXAMPLES, 

4.  For  what  time  must  a  force  of  60  grams  (gravitation  meas- 
ure) act  on  a  body  weighing  2  kilos  to  give  it  a  velocity  of  6 
metres  per  second  ? 

5.  A  body  weighing  140  kilos  is  moved  by  a  constant  force, 
which  generates  a  velocity  of  2  meters  per  second  in  one  second : 
What  weight  could  the  force  support? 

6.  What  constant  force  (a)  in  gravitation  measure  (grams),  (b) 
in  absolute  measure  (dynes),  will  cause  a  body  weighing  490 
grams  to  pass  over  400  meters  in  10  seconds  on  a  smooth  hori- 
zontal surface? 

7.  A  constant  force  of  980  dynes  gives  a  body  an  acceleration 
of  7  meters  per  second  in  one  second :  What  is  the  weight  of  the 
body  (in  grams)? 

8.  What  weight  could  a  force  equal  to  1  dyne  support? 

XV.  Centripetal  and  Centrifugal  Forces.   Articles  77-81 ;  pp.  83-8. 

1.  A  ball  weighing  10  kilos  is  whirled  by  means  of  a  string 
around  a  centre  at  a  radius  of  2  meters,  with  a  linear  velocity  of 
7  meters  per  second :  What  is  the  value  of  /,  and  what  is  the 
tension  of  the  string  (i^)? 

2.  A  ball  weighing  14  kilos  attached  to  a  centre  at  a  distance 
of  3  meters  makes  420  revolutions  in  a  minute  {it  =  3|):  What  is 
the  pull  on  the  centre? 

XVI.  Friction,    Articles  82-94;  pp.  90-98. 

1.  A  fo.rce  of  6  kilos  is  just  sufficient  to  move  a  body  weighing 
48  kilos  uniformly  along  a  horizontal  plane:  What  is  the  coeffi- 
cient of  friction? 

2.  The  value  of  //  is  .3,  the  weight  of  the  body  is  16  kilos: 
What  force  is  required  to  move  it  uniformly? 

3.  It  is  found  that  a  force  of  40  grams  suffices  to  move  a  body 
uniformly  on  a  horizontal  surface,  where  the  value  of  the  co- 
efficient of  friction  is  known  to  be  .25:  What  is  the  weight  of  the 
body? 

4.  A  body  weighing  15  kilos  is  just  on  the  point  of  sliding 
when  the  surface  it  rests  upon  is  inclined  20° :  {a)  What  is  the  co- 
efficient of  friction  and  the  force  of  friction?  {b)  If  the  weight  of 
the  body  is  doubled,  what  values  have  these  quantities? 


INTEODUCING  THE  METRIC   UNITS.  271 

5.  A  body  weighing  13  kilos  rests  on  an  inclined  plane  whose 
angle  of  inclination  is  30°  and  where  /n  =  .Q:  What  is  the  force 
of  friction? 

XVII.    Woi±    Articles  95-100;  pp.  101-105. 

[The  UNIT  OP  WORK  is  usually  taken  as  one  kilogram-meter;  on 
the  C.  G.  S.  system  it  is  an  erg,  or  one  dyncrcentimeter.'] 

1.  How  many  foot-pounds  correspond  to  one  kilogram-meter? 

2.  A  weight  of  300  kilos  is  raised  to  the  top  of  an  inclined 
plane  whose  length  is  1200  metres,  and  the  angle  of  inclina- 
tion =  10°:  What  work  is  done? 

3.  A  sled  weighing  600  kilos  is  dragged  15  kilometers  on  the 
snow,  where  the  coefficient  of  friction  is  .075:  What  work  is 
done  against  friction  ? 

4.  How  much  work  is  done  against  friction  in  dragging  a 
weight  of  200  kilos  a  distance  of  1000  meters  along  a  horizontal 
plane,  if  the  coefficient  of  friction  is  .5? 

5.  A  weight  of  100  kilos  is  dragged  up  an  inclined  plane  whose 
length  is  2600  meters,  and  whose  height  is  1000  meters  (yu  =  .3): 
How  much  work  is  done? 

XVIII.  Potential  and  Kinetic  Energy.     Articles  101-118; 
pp.  106-124. 

1.  How  many  kilogram-meters  of  work  are  equivalent  to  one 
heat-unit  {i.e.,  to  raise  1  kilo  of  water  1°  C.)? 

2.  The  weights  of  a  clock  weigh  20  kilos,  and  they  have  10 
meters  to  fall  •  How  much  work  do  they  represent  when  wound 
up  ? 

3.  A  mill-pond  has  a  surface  of  1000  square  meters  and  an 
average  depth  of  1  meter;  supposing  it  50  meters  above  the  sea- 
level,  how  much  potential  energy  does  it  represent? 

4.  How  much  work  is  accumulated  or  stored  up  (=  kinetic 
energy)  in  a  cannon-ball  weighing  100  kilos  and  moving  at  a  rate 
of  280  meters  per  second?  How  much  heat  will  be  generated  if 
its  mass  motion  is  entirely  destroyed  by  the  impact  with  the 
target? 

5.  A  bullet  weighing  30  grams  has  a  velocity  of  420  meters  peif 
second:  How  much  woik  can  it  do? 


272  ADDITIONAL  EXAMPLES, 

6.  A  body  weighing  12  kilos  is  projected  along  a  rough  hori- 
zontal plane  (jj.  =  .25)  with  an  initial  velocity  of  140  meters 
per  second:  How  far  {a)  will  it  go  before  coming  to  rest,  and 
how  long  (b)  will  it  slide? 

7.  A  hammer  weighing  5  kilos  and  moving  with  a  velocity  of 
1.4  meters  per  second  drives  a  nail  into  a  plank  1  centimeter: 
What  resistance  does  it  overcome  ? 

8.  A  weight  of  500  kilos,  used  as  a  pile-driver,  falls  6  meters 
and  drives  the  pile  in  2  centimeters:  What  resistance  does  it 
overcome? 

XIX.  Parallelogram  of  Forces.    Articles  124-136;  pp.  129-142. 

1.  Two  forces,  P  =  70  grams,  Q  =  240  grams,  act  at  right 
angles  to  each  other :  Eequired  the  magnitude  and  the  direction 
of  their  resultant. 

2.  Two  forces,  P  =  12  kilos,  ^  =  10  kilos,  act  at  an  angle  of 
120° :  Required  the  magnitude  of  B. 

3.  Of  two  forces,  P  =  12  and  ^  =  24  kilos,  the  angle  between 
Q  and  B  is  30°:  Required  B  and  the  angle  between  P  and  Q. 

4.  A  peg  in  a  wall  is  pulled  by  two  strings  with  forces  of  4 
kilos  each;  they  are  equally  inclined  downward  (40°)  to  the 
vertical:  What  weight  hung  on  the  peg  would  give  an  equal 
strain? 

5.  A  peg  in  a  wall  is  pulled  by  two  strings,  one  horizontal  with 
a  tension  of  350  grams,  and  the  other  vertical  with  a  tension  of 
840  grams:  What  single  force  would  exert  an  equal  pull  upon  it? 

6.  A  weight  is  supported  by  two  equal  strings  attached  to  nails 
in  the  ceiling  and  enclosing  an  angle  of  120°;  the  tension  of  each 
string  is  8  kilos:  What  is  the  weight  supported? 

XX.  Besoluiion  of  Forces.    Articles  137,  138;  pp.  143-146. 

1.  A  force  of  60  kilos  is  exerted  in  a  direction  N.  20°  E. :  What 
portion  of  it  is  felt  nortfi?  what  portion  east? 

2.  A  weight  of  12  kilos  is  supported  by  two  strings  at  an  angle 
of  120° ;  one  (a)  goes  (Fig.  66,  p.  144)  horizontally  to  the  vertical 
wall,  and  the  other  (p)  to  the  ceiling:  What  is  the  tension  of  the 
two  strings? 

8.  A  picture,  whose  weight  is  40  kilos,  is  supported  by  a  cord 


INTRODUCING  THE  METRIC   UNITS.  273 

attached  to  the  upper  corners  and  carried  over  a  nail  so  as  to 
include  an  angle  of  75°.  If  the  top  of  the  picture  is  horizontal, 
what  are  the  tensions  of  the  strings? 

4.  A  horse  drags  a  sled  by  a  rope  inclined  at  the  ground  at  an 
angle  of  10°;  the  tension  of  the  rope  is  300  kilos:  What  is  the 
effective  component  of  the  force  exerted? 

5.  A  weight  of  640  grams  is  supported  by  two  strings,  one  of 
which  makes  an  angle  of  30°  with  the  vertical,  and  the  other  60° : 
Find  the  tension  of  each  string. 

XXI.    Resolution  of  Forces  along  two  Rectangular  Axes. 
Articles  140,  141,  147-149;  pp.  147-149,  156-158. 

1.  Find  the  magnitude  and  direction  of  the  resultant  of  the 
following  forces:  P  =  100  kilos,  Q=z50,  S  =  200.  The  angle  be- 
tween P  and  Q  =  60°,  between  Q  and  8  =  270°. 

2.  Kequired  the  magnitude  and  direction  of  the  resultant  of  the 
following  forces:  P=  Q  =  S  =  T  =  100  kilos.  The  angles  are 
as  follows:  between  P  and  ^  =  60°,  between  Q  and  8  =  120°, 

.between  5 and  r=120°. 

XXII.  Parallel  Forces.    Articles  143-149;  pp.  152-159. 

1.  A  rigid  rod,  supported  at  the  ends  A  and  B,  has  a  weight  of 
24  kilos  hung  2  meters  from  A  and  4  meters  from  B:  What  pres- 
sures do  the  supports  feel?  The  weight  of  the  rod  itself  is  neg- 
lected here,  as,  too,  in  the  following  examples. 

2.  ABG  is  a  rigid  rod;  at  P  a  weight  TTis  hung,  so  that  AB  = 
24  and  BG  =  32 ;  the  pressure  at  A  is  14  kilos :  What  is  the  pres- 
sure at  G,  and  what  is  W  ? 

3.  A  weight  of  80  kilos  is  carried  by  means  of  a  rigid  rod  on 
the  shoulders  (at  the  same  height)  of  two  men  A  and  P;  the  dis- 
tances from  them  are  2  and  3  meters  respectively :  What  weight 
does  each  carry? 

4.  A  table  has  as  its  top  an  equilateral  triangle  J.PC  (Fig.  82) ; 
a  weight  of  10  kilos  is  placed  at  0,  so  that  the  perpendicular  dis- 
tance from  0  on  BG  =  18  centimeters,  and  those  on  AG,  AB 
each  equal  36  centimeters:  What  is  the  pressure  on  each  of  the 
three  legs? 

5    A  rod  40  centimeters  long  and  whose  weight  acts  at  ita 


274 

middle  point  rests  on  two  vertical  props  placed  at  the  ends: 
Where  must  a  weight,  equal  to  twice  that  of  the  rod,  be  placed 
that  the  pressure  on  the  props  shall  be  as  4  :  1  ? 

XXIII.  Moments.     Articles  151-156 ;  pp.  161-167. 

1.  A  force,  P  =  24  kilos,  acts  atright  angles  to  an  arm  3  meters 
long:  What  is  its  moment? 

2.  A  rigid  rod  AB,  2  meters  long  and  free  to  turn  about  B,  is 
acted  on  by  a  force,  P  =  30  kilos,  whose  direction  makes  an  angle 
of  30°  with  AB:  What  is  the  moment  of  P  ?  If  the  angle  is  150", 
what  is  the  moment? 

3.  A  force,  P  =  100  kilos,  acts  at  the  extremity  of  a  rod,  AB, 
4  meters  long,  and  at  an  angle  of  120° :  What  is  the  moment  of  P 
about  B ? 

4.  A  bar  5  meters  long  and  pivoted  at  the  middle  has  a  weight 
of  10  kilos  hung  at  one  extremity:  What  is  the  moment  of  the 
weight  (a)  when  the  bar  is  horizontal,  (b)  when  it  makes  an  angle 
of  20°  below,  and  (c)  of  70°  above  with  the  horizontal  position? 

XXIV.,  XXV.   Centre  of  Gramty— Stability.     Articles  159-177; 
pp.  169-189. 

1.  Where  is  the  centre  of  gravity  of  two  bodies,  A  and  B, 
weighing  30  and  42  grams  respectively,  rigidly  connected  by  a 
weightless  rod  36  centimeters  long? 

2.  A  rod  AB,  1  meter  long  and  weighing  250  grams,  has  a 
weight  P  =  4|  kilos  hung  at  the  end  B  :  Where  will  it  balance? 

3.  What  weight  must  be  hung  at  the  end  of  a  rod  i  meter  long 
and  weighing  30  grams  that  it  may  balance  25  millimeters  from 
that  end? 

4.  A  rod  2  meters  long  and  having  a  weight  of  5  kilos  at  one 
end  balances  at  a  point  2  millimeters  from  this  end:  What  is  its 
weight? 

5.  A  uniform  rod  AB,  1  meter  long  and  weighing  1  kilo,  has  a 
weight  of  750  grams  at  the  end  A,  and  one  of  250  at  the  end  B: 
Where  will  it  balance? 

6.  A  uniform  metal  wire  is  bent  into  the  form  of  an  isosceles 
triangle,  ABC,  so  that  AB=AG=  117  millimeters,  and  BC=  90 
millimeters:  Where  is  the  centre  of  gravity? 


INTRODUCING  THE  METRIC  UNITS.  275 

7.  A  table  2  meters  square  stands  upon  four  legs,  each  of  which 
is  300  millimeters  in  from  the  adjacent  edges;  its  height  is  800 
millimeters  and  its  weight  24  kilos:  What  is  the  least  force 
required  to  put  it  on  the  point  of  overturning  if  applied  at  the 
edge  (a)  as  a  horizontal  push?  (b)  as  a  pressure  directly  down? 

8.  A  table,  having  a  circular  top  of  i  meter  radius,  is  sup- 
ported on  four  legs  placed  at  the  edge  and  at  equal  distances 
from  one  another;  the  height  is  f  meter  and  the  weight  16  kilos: 
What  is  the  least  force  that  will  put  it  on  the  point  of  turning  it 
if  applied  at  the  top  (a)  as  a  horizontal  push?  (b)  as  a  pressure 
down?  (c)  acting  vertically  upward? 

9.  What  work  would  be  done  in  overturning  a  cylindrical 
column  of  stone  weighing  10,000  kilos,  3  meters  high  and  1  meter 
in  diameter,  supposing  that  the  centre  of  gravity  is  on  the  axis 
(a)  at  the  middle?  (b)  .25  meter  from  bottom,  (c)  the  same  distance 
from  top? 

XXVI.  Lever.    Articles  185-190;  pp.  195-201. 

[The  weight  of  the  lever  is  to  be  neglected,  except  when  other- 
wise stated.] 

1.  The  force  P=  40  kilos  acts  as  in  Fig.  126,  p.  196;  AF  = 
2  meters  and  AB  =  3i  meters:  What  weight  can  be  supported? 

2.  If  (Fig.  127,  p.  196)  AB=5,BF=2i  meters,  and  the  weight 
is  60  kilos,  what  force  Pis  required  to  support  it? 

3.  If  (Fig.  128,  p.  196)  AB=  14,  AF=2  meters,  and  P  =  24 
kilos,  what  weight  can  P  support? 

4.  AFC  is  a  bent  lever  (Fig.  129,  p.  197);  AF=  12,  FG=-U, 
AFG=135°,  P=  20  kilos:  What  is  TF? 

5.  CFB  is  a  bent  lever  (Fig.  131,  p.  197);  CF=  16,  PJD  =  24, 
PI)F=  150°,  FCW=  120°,  and  TF=  60  kilos:  What  is  P  ? 

6.  A  heavy  uniform  rod  BF  (Fig.  133,  p.  199),  weighing  10 
khos  and  f  meter  long,  is  hinged  at  P;  it  is  supported  by  a 
string  carried  from  i)  to  a  point  E,  250  milimeters  vertically 
above  F:  What  is  the  tension  of  the  string? 

7.  A  heavy  uniform  shelf  BF  (Fig.  134),  ^  meter  wide,  weigh- 
ing 24  kilos,  and  hinged  at  F,  is  supported  by  a  prop  carried 
from  C'  (CP=  400  millimeters)  to  a  point  F  below  F,  so  that 
CF=  FE:  What  pressure  does  this  prop  feel? 


276  ADDITIONAL   EXAMPLES, 

8.  Forces  of  8  and  12  kilos  act  at  the  extremities  of  a  straiglit 
bar  4  meters  long,  and  in  directions  making  angles  of  120°  and 
150°  respectively  with  it:  "Where  is  the  fulcrum  in  case  of  equi- 
librium? 

XXVII.,  XXVIII.  Balance— SUelyard.    Articles  191-196; 
pp.  202-210. 

1.  A  body  is  equivalent  to  a  weight  of  6  kilos  in  one  pan  of  a 
false  balance,  and  of  6^  kilos  in  the  other:  What  is  the  true 
weight? 

2.  A  body  is  equivalent  to  a  weight  of  84  grams  from  one  arm 
of  a  false  balance,  and  of  72  grams  from  the  other :  What  is  the 
ratio  of  the  lengths  of  the  arms? 

3.  The  true  weight  of  a  body  is  1  gram,  its  apparent  weight  in 
one  pan  of  a  balance  is  960  milligrams:  What  would  it  seem  to 
weigh  in  the  other  pan? 


4.  The  whole  length  of  a  steelyard  is  f  meter:  CG  (Fig.  138)  =  6 
millimeters,  CA  =  20 millimeters,  P=  250 grams,  and  Q  =  i kilo: 
(a)  Where  is  the  zero  of  the  scale?  (b)  What  is  the  length  of 
graduation  for  1  kilo?    (c)  How  large  weights  can  it  be  used  for? 

5.  The  weight  of  the  beam  of  a  steelyard  is  1  kilo,  and  the 
distance  of  its  centre  of  gravity  is  16  millimeters  from  the  ful- 
crum: Where  must  a  counterpoise  of  640  grams  be  placed  to 
balance  it? 

6.  The  length  of  a  Danish  steelyard  is  i  meter,  its  weight  is 
1^  kilos,  and  it  acts  at  a  point  50  millimeters  from  one  end ;  a 
body  weighing  6  kilos  hangs  at  the  other  end:  Where  is  the 
fulcrum? 

XXIX.,   XXX.     Wheel  and  AxU  —  PuUey.     Articles   20:;^-226; 
pp.  216-234. 

1.  The  radius  of  the  axle  is  50  millimeters,  that  of  the  wheel 
is  f  meter,  and  the  power  acting  is  60  kilos:  What  weight  is 
supported? 

2.  A  man,  exerting  a  force  of  40  kilos  on  a  lever-arm  IJ  meters 
long,  turns  a  capstan ;  the  radius  of  the  circle  about  which  the 
rope  is  wound  is  150  millimeters:  What  is  the  pull  fell  upon  the 
anchor? 


INTKODUCING  THE   METEIC   UlS^ITS.  271 

3.  A  weight  of  300  kilos  hangs  by  a  rope  20  millimeters  ia 
thickness;  ?•  =  .2  meter,  and  B  =  1.25  meters;  the  power  acts  on 
a  lever-arm  without  a  rope :  What  is  P  ? 

4.  A  power  of  40  kilos  balances  a  weight  of  600  kilos;  the 
radius  of  the  axle  is  75  millimeters:  What  is  the  diameter  of  the 
wheel? 


5.  In  a  combination  of  pulleys,  as  in  Fig.  162,  W=  704  kilos, 
and  P  =  44  kilos:  How  many  pulleys  are  there? 

6.  In  a  combination  of  pulleys,  as  in  Fig.  163,  W=  288  kilos 
and  P=  48  kilos:  How  many  pulleys  are  there? 

7.  In  a  combination  as  in  Fig.  164,  W=  837  kilos,  P=  27 
kilos:  What  is  the  number  of  pulleys? 

8.  In  the  single  movable  pulley,  1^=100  kilos:  Calculate  the 
value  of  P  when  2a  =  45°,  =  135°. 

9.  What  force  is  needed  to  support  200  kilos  by  the  second  sys- 
tem of  pulleys,  there  being  4  in  all?  What  is  the  force  if  each 
pulley  weighs  i  kilo? 

XXXI.  Inclined  Plane.    Articles  227-232;  pp.  235-241. 
[The  plane  is  supposed  to  be  perfectly  smooth.] 
'    1.  The  angle  of  the  plane  is  30°,  the  weight  is  120  kilos:  What 
force  is  required  to  support  the  weight  (a)  acting  parallel  to  the 
plane?  (b)  acting  horizontally?  (c)  acting  at  an  angle  of  60°  with 
the  plane? 

2.  What  is  the  reaction  of  the  plane  in  the  three  cases  in  ex- 
ample 1? 

3.  A  force  of  60  kilos  acts  parallel  to  an  inclined  plane :  What 
weight  can  it  support  in  the  following  cases — the  angle  of  the 
plane  is  (a)  20°,  (b)  40°. 

4.  A  weight  of  60  kilos  rests  on  an  inclined  plane :  What  force 
acting  parallel  to  the  plane  is  required  to  support  it,  if  the  angle 
of  the  plane  has  the  same  values  as  in  example  3? 

5.  A  force  of  50  kilos  acts  horizontally  to  an  inclined  plane: 
What  weight  can  it  support  if  the  angle  of  the  plane  is  45°? 

6.  If  a  horse  can  raise  800  kilos  vertically,  what  weight  can  he 
raise  on  a  railway  having  a  grade  of  44  feet  to  the  mile? 


278  ADDITIONAL  EXAMPLES. 

XXXII.,  XXXIII.  Wedge— Screw.    Articles  233-242 ;  pp.  243-250. 

1.  A  wedge  is  isosceles  in  shape  and  has  an  angle  of  30° ;  if 
J*=  20  kilos,  what  is  the  resistance  on  each  face? 

2.  A  wedge  is  isosceles,  and  the  angle  60° ;  a  force  of  100  kilos 
acts  at  the  back:  What  are  the  other  two  forces? 

3.  A  weight  is  isosceles,  the  power  acting  on  the  back  is  20 
kilos,  and  the  forces  on  the  other  sides  are  60  kilos  each:  What 
is  the  angle  of  the  wedge? 

4.  The  wedge  is  right-angled  and  the  three  sides  have  lengths 
of  25,  20,  15  (back):  If  P=  100,  what  are  the  other  two  forces? 


5.  What  weight  can  be  raised  by  a  power  of  25  kilos  acting  on 
a  lever-arm  |  meter  long,  if  the  screw  has  2  threads  to  the  centi- 
meter? 

6.  If  a  power  of  40  kilos  acting  on  an  arm  1  meter  long  can 
support  a  weight  of  8000  kilos,  what  will  be  the  distance  between 
the  threads  of  the  screw? 

7.  A  screw  has  1  thread  to  the  centimeter;  the  circumference 
described  by  the  power  is  2  meters :  What  power  is  needed  to 
support  a  weight  of  3000  kilos? 

8.  While  the  point  of  application  of  the  power  makes  a  revo- 
lution of  1  meter,  the  screw  advances  6  millimeters ;  the  power  is 
50  kilos:  What  is  the  weight  raised? 

XXXIV.  Pendulum.     Articles  244-247;  pp.  252-261. 

[The  length  of  the  seconds  pendulum  at  New  York  is  about 
.9932  meter.] 

1.  Required  the  length  of  a  pendulum  at  New  York  which 
will  vibrate  (a)  in  ^  second,  (b)  in  3  seconds,  (c)  in  li  seconds. 

2.  Required  the  length  of  the  seconds  pendulum  where  the 
acceleration  of  gravity  is  9.83  (that  is,  near  the  pole). 

3.  Required  the  length  of  a  pendulum  to  vibrate  in  2  seconds, 
where  the  value  of  g  is  9.78. 

4.  How  many  beats  in  a  minute  would  a  pendulum  H  meters 
long  make  in  New  York? 


ANSWERS  TO  EXAMPLES. 


Pages  12,  13.     I.    Uniform  Motion  of  Translation  or  Rotation. 
Articles  17-21. 

(1)490.91  miles.  (2)  44  feet  per  second.  (3a)  5|f  feet  per 
second;  (36)  117^  yards  per  minute.  {A.a)  3 miles;  (4&)  25  seconds. 
(5)  1  mile.     (6)  7.39  miles.    (7)  19.01  miles. 

(8)  1047.2  miles  per  hour,  or  1535.9  feet  per  second.  (9)  523.6 
miles  per  hour,  or  767.9  feet  per  second.  (11)  .000073.  (12a) 
|7r  =  4.189;  (126)  12.57  feet  per  second.  (13a)  628.3;  (136)  8567.9 
miles.     (14a)  5.236  feet  per  second;  (146)  10.472;  (14c)  26.18. 

Pages  21,  22.     II.  Uniformly  Accelerated  Motion.     Articles  22-28. 
A.  Falling  Bodies. 

(la)  480  ft.  per  sec. ;  (16)  3600  ft. ;  (Ic)  464  ft. ;  (1^  1296  ft. 
(2a)  18  sec. ;  (26)  576  ft.  per  sec.  (3a)  16  sec. ;  (36)  4096  ft.  (4a) 
11  sec. ;  (46)  1936  ft.  (5a)  12  sec. ;  (56)  2304  ft.  (6)  i  :  i  :  1  :  3  :  f ; 
the  velocities  are  8, 16,  33,  96,  144  ft.  per  sec.  (7)  yV  =  i  =  1  :  9  :  -«^; 
the  distances  are  1  ft.,  4,  16,  144,  324  feet.  (8a)  31.46  sec;  (86) 
1006.86  ft.  (9a)  55^  ft.;  (96)  108f.  (10)  100  ft.  (11a)  192  ft., 
256  ft.,  384  ft. ;  (116)  when  B  has  fallen  5i  sec.  (12)  3.94  sec. 
(13)  576  ft. 

B.  General  Case. — Acceleration  =/. 

(1)  8  ft.-per-sec.  per  sec.  (2a)  20;  (26)  640  ft. ;  (2c)  160  ft.  per 
sec.  (3a)  72  ft.  per  sec. ;  (36)  216  ft.  (4)  8.  (5a)  249.6  ft.  per  sec. ; 
(56)  374.4  ft.  (6)  7i  sec.  (7)  6  sec.  and  96  ft.  (8)  For  the  earth 
v  =  32,  64,  96;  and  s  =  16,  48,  80.  The  corresponding  values  for 
the  sun  are  28  times  greater.  (9)  Yes.  (10)  Three  times  as  far, 
i.e.,  150  ft. 


280  ANSWERS   TO   EXAMPLES. 

Pages  30,  31.    III.  Composition  of  Velocities.    Articles  29-37. 

(la)  1  mile  per  hour;  (16)  9  miles;  (Ic)  4  miles;  (IcZ)  12  miles. 
(2a)  At  an  angle  of  38°  40'  with  the  line  drawn  directly  across 
{BAG,  Fig.  11);  (25)  6.4  miles  per  hour;  (2c)  24  minutes;  {M)  If 
miles  down  stream  {BG,  Fig.  11).  (3a)  53°  8'  up  stream  {BAG, 
Fig.  12);  (3Z>)  3  miles  per  hour;  (3c)  40  minutes.  (4a)  At  an  angle 
of  73°  51'  up  stream  made  with  the  line  drawn  directly  across; 
(46)1.606  miles  per  hour;  (4c)  1  hr.  26.3  min.  (5a)  13°  51'  up 
stream  measured  from  the  line  directly  across;  (56)  5.61  miles  per 
hour;  (5c)  24.7  min.  (6)  Three  fourths  of  a  mile  down  stream. 
(7a)  13  ft.  per  sec. ;  (76)  N.  22°  37'  E.  (8a)  N.  22°  30'  E. ;  (86) 
14.78  ft.  per  sec.  (9a)  15  ft.  per  sec;  (96)  36°  52'  with  his  own 
direction.  (10)  0.  (11)  2.07  miles  per  hour  due  west.  (12a)  128.06 
yds.  per  minute  at  an  angle  38°  40'  down  stream;  (126)  960  yards 
below  the  starting-point  {BG,  Fig.  11);  (12c)  53°  8'  up  stream 
{BAG,  Fig.  12);  {\M)  20  minutee. 

Pages  31,  32.     IV.  Mesolutlon  of  Gonstant  Velocities.     Article  38, 

(1)  7.79  ft.  per  sec.  (2)  6,  10.39,  12,  10.39,  0.  (3)  6.4  ft.  per 
sec.  parallel  AG,  and  4.8  parallel  AB.  (4)  5.2  miles  per  hour 
north,  and  3  miles  east.  (5a)  8.66;  (56)  5.  (6)  100  yds.  per  min. 
and  at  an  angle  of  53°  8'  up-stream.  (7)  11.05  ft.  per  sec, 
a  =  33°  34'. 

Pages  33,  34.     V.  Falling  down  an  Inclined  Plane.     Article  40. 

(la)  16  ft.-per-sec  per  sec;  (16)  128  ft.;  (Ic)  64  ft.  per  sec; 
{Id)  56  ft.  (2a)  10  seconds;  (26)  80  ft.  per  sec.  (3)  3.57  sec. 
(4a)  2  ft.-per-sec.  per  sec  ;  (46)  36  feet.  (5a)  4  ft.-per-sec.  per  sec  ; 
(56)  784  feet.  (6a)  1024  feet;  (66)  128  ft.  per  sec.  (7)  The  times 
are  5,  7i,  10,  15,  20  sec ;  the  acquired  velocity  is  160  ft.  per  sec 
in  all. 

Pages  39,  40.     VI.  Bodies  jn^qjected  mrtically  downward.    Articles 
41,  42. 

(la)  260  ft.  per  sec. ;  (16)  1036  feet;  (Ic)  244  feet.  (2a)  5i  sec; 
(26)  196  ft.  per  sec.  (3)  42  ft.  per  sec.  (4)  75  ft.  per  sec.  (5a) 
55  ft.  per  sec. ;  (56)  675  feet.  (6)  21  ft.  per  sec.  (7a)  64  feet; 
(76)3iscc.     (8a)  1216  feet;  (86)  400  feet. 


ANSWEKS   TO  EXAMPLES.  281 

Pages  40,  41.    VII.  Bodies  projected  vertically  upward.    Article  44. 

{\a,  lb)  9  sec;  (Ic)  1296  feet;  {Id)  273  feet  and  16  feet.     (2a)  3 

or  9  sec. ;  (25)  15  or  -  3  sec.     (3a)  ^  sec. ;  (35)  8^  sec. ;  (3c)  15  sec. 

(4)  240  ft.  per  sec.  (5)  12  sec.  (6)  192  ft.  per  sec.  (7a)  112  ft. 
per  sec. ;  (76)  same  as  (7a).  (8)  256  ft.  per  sec.  (9)  4  feet  from 
the  top  after  i  second.  (10a)  440  feet ;  (105)  88  feet;  (10c)  88  feet. 
(11)  Two  seconds  after  the  second  ball  started.  (13)  It  will 
actually  ascend  3  seconds  (though  apparently  falling  as  seen  from 
the  balloon)  and  then  descend,  reaching  the  ground  after  7 
seconds  longer,  or  10  seconds  in  all;  the  total  distance,  up  and 
down,  is  928  feet. 

Pages  41,  42.   VIII.  Projected  up  or  down  a  smooth  Inclined  ^lane. 
Article  45. 
(la)  8  sec. ;  (15)  89  feet,  per  sec.     (2a)  96  ft.  per  sec. ;  (25)  12 
sec.     (3a)  109  ft.  per  sec. ;  (35)  308  feet;  (3c)  101  feet.     (4a)  5  sec. ; 
(45)  200  feet;  (4c)  48  ft.  per  sec. ;  {^d)  -  48. 

Page  42.  IX.  Bodies  projected  against  Friction.  Articles  41,  42. 
(la)  12  ft.-per-sec.  per  sec. ;  (15)  after  10  sec. ;  (Ic)  600  ft.  (2a) 
5  sec. ;  (25)  100  feet;  (2c)  16  ft.  per  sec.  (3)  22  seconds  and  242 
feet.  (4a)  60  miles  per  hour;  (45)  four  times  as  far.  (5)  4000 
feet;  20  seconds. 

Page  50.  X.  Projectiles.  Articles  47-51. 
(la)  5  seconds;  (15)  692.8  feet;  (Ic)  100  feet.  (2)  2  or  3  seconds. 
(3)  7°  14'  or  82°  46'.  (4)  367.65  ft.  per  sec.  (5)  2\  seconds  at  a 
distance  of  3000  feet.  (6a)  1056  ft.  per  sec. ;  (65)  400  feet.  (7a) 
120  feet;  (75)  48  and  80  ft.  per  sec.  (8c)  after  1  second;  {M)  66 
feet  in  a  horizontal  line  from  the  point  where  it  was  dropped. 

Page  55.     XI.  Mass — Density —  Volume.     Article  56. 
(1)  1  :  4^.   (2)  lir  :  1.    (3)  9  :  8.   (4)  2|  :  1.    (5)  .49  lb.  (6)  1.21  :  1. 
(7)  1.1.    (8)  1.024. 

Page  63.    XII.  Force  of  Gravity.    Articles  63-65. 
(1)  4  V2  times  the  earth's  radius.     (2)  g'  (sun)  :  g  (earth)  = 

?^^  :  1;    that  is,  g'  =  27.9  g.    (3)  About  ^  of  g.    (4)  317  :  1. 

(5)  32.047  (only  the  difference  in  distance  from  the  earth's  centre 
is  considered). 


282  ANSWERS   TO   EXAMPLES. 

Page  72.    XIII.  Collision  of  Inelastic  Bodies.    Article  70. 

(1)  13^  ft.  per  sec.  (3)  8  ft.  per  sec.  (3)  20  ft.  per  sec.  (4)  Re- 
spectively 8,  6,  4f.  (5)  4:1.  (6)  8  :  7.  (7)  7  :  5.  (8)  :9|  ft.  per 
sec.     (9)  1631.76  ft.  per  sec. 

Pages  80-82.     XIV.  General  Dynamical  Problems.     Articles  68 
and  73-76. 

{la)  4  ft.-per-sec.  per  sec. ;  (15)  8  feet.  (2a)  At  a  distance  equal 
to  the  earth's  radius ;  (2J)  one  fourth  as  great  as  at  the  surface. 
(3)  5i  ft.-per-sec.  per  second;  the  tension  {T)  =  10  lbs.  (4)  -^  of 
g.  {5a)  100  lbs. ;  {5b)  112.5.  (6)  20  seconds.  (7)  9|  feet.  (8)  1^ 
oz.  (9)  j\  lb.  {10a)  27  lbs.;  (106)  21  lbs.  (11)  If  the  velocity  is 
gained  in  1  second,  40  lbs.  (12a)  lOf  ft.-per-sec.  per  second; 
(126)  85i  feet.  (13)  /=  20;  15  oz.  =  30  poundals.  (14)  l^f  lbs. 
(15)  13f  sec.    (16)  300  lbs.     (17)  96  lbs. 

(18a)  8  sec;  (186)  256  feet.  (19)  5  lbs.  (20a)  666|  feet. ;  (206) 
8i  sec.    (21a)  215.4  ft.  per  sec. ;  (216)  9.28  sec. 

Pages  88,  89.     XV.  Centripetal  and    Centrifugal  Foi'ces.     Arti- 
cles 77-81. 

(1)  /=  112;  tension  =  70  lbs.  (2a)  They  are  increased  4  times; 
(26)  they  are  diminished  one  half.  (3)  986.97  lbs.  (4)  16  ft.  per 
sec.  (5)  2.  (6)  .726  ton.  (7)  74.02.  lbs.  (8)  16.2  feet  (only  the 
tension  caused  by  the  circular  motion  is  considered).     (9)  1  foot. 

Pages  99,  100.    XVI.    Friction.    Articles  82-94 

(l)j«  =  .25.  (2)  4.8  lbs.  (3)  28  lbs.  (4a)  // =  .364,  ir=  5.13 
lbs.;  (46)  n  =  .364,  F=  10.26.  (5)  4.657  lbs.  (6c)  9.6  and  12  lbs. 
(7)  //  =  .54.  (8a)  6  lbs. ;  (86)  5.638  lbs.  (9a)  P=Wsina-F  = 
8.8  lbs.;  (96)  P  =  Tf  sin  a  +  i^'  =  15.2  lbs.  (10)  10.35  lbs. 
(11)  13.05  lbs.     (12)208.46.     (13)167.42.     (14)  6f  lbs.    (15)  12  lbs. 

Pages  105,  106.    XVII.  Work.    Articles  95-100. 

(1)  125,028  ft.lbs.  (2)  102,492.7  ft. lbs.  (3)  1,440,000  ft.lbs.  (4) 
12,672,000  ft.lbs.  (5)  5,940,000  ft.lbs.  (6)  600,000  ft.lbs.  (7) 
Against  gravity  250,000  ft.lbs.,  against  friction  180,000  ft.lbs.; 
total  430,000  ft.lbs.     (8)  Against  gravity  60,000  ftlbs.,  against 


ANSWERS  TO  EXAMPLES.  283 

friction  24,000   (horizontal    surface),  20,784.6  (inclined  plane); 
total  104,784.6  ft.lbs. 

Pages  124-126.     XVIII.  Potential  and  Kinetic  Energy. 
Articles  101-108. 

(1)  1800  ft.lbs.  (2)  3,267,000,000  ft.lbs.  (3)  9,375,000  ft.lbs.  per 
minute,  or  284.09  liorse-power.  (4)  250,000  ft.lbs.  per  minute, 
or  7.58  horse-power.  (5)  6924.46  heat-units,  and  0.111°  C.  (6) 
4,500,000  ft.lbs.,  3237.41  heat-units.  (7)  625  ft.lbs.  (8a)  10,000 
feet;  (86)  four  times  as  far.  (9)  Same  distance.  (10«)  6400  feet; 
(10&)  40  sec;  (10c)  624  feet;  {IM )  304  ft.  per  sec.  (11a)  5000  feet; 
(116)50  sec;  (lie)  18,000  ft.lbs.  (12)  60.47  feet.  (13)/  (the  re- 
tardation on  the  plane)  =  1(32);  (13a)  2000  feet;  (136)  12i  sec. 
(14)  343.16  ft.  per  sec.  (the  weight  of  the  body  does  not  enter  into 
tlie  problem.  (15a)  1360  X  Tf  ft.lbs. ;  (156)  6800  feet.  (16)i2  = 
721b6.    (17)  240,000  lbs.    (18)  37,500  lbs. 

Pages  142, 143.    XIX.  Parallelogram  of  Foi-ces.    Articles  126-136. 

(1)  25  lbs.,  a  =  73°  44'.  (2)  9.097,  a  =  30°  59'.  (3)  Q  =  16.16, 
r  =  83°  29'.  (4)  /3  =  64°  39',  r  =  114°  9',  P  =  5.94;  or  /3  = 
16°  21',  r  =  65°  51'.  P  =  1.85.  (5)  r  =  120°,  B  =  27.71  lbs.  (6) 
12.26  lbs.  (7)  35  lbs.,  inclined  53°  8'  to  the  horizontal.  (8)  20.78 
lbs.  (9)  15  and  20  lbs.  (10)  100  lbs.,  16°  16' and  73°  44'.  (11)23.43 
oz.     (12)  5.72  and  11.44  lbs.    (13)  41.57  lbs. 

Pages  150,  151.    XX.  Besolution  of  Forces.    Articles  137,  138. 

(1)  106.05  lbs.  N.  and  the  same  E.  (2)  5  lbs.,  5.18,  5.77,  7.07, 10, 
19.30,  00 .  (3)  a  =  23.83,  6  =  31.11  lbs.  (4)  a  =  34.64  lbs.,  6  =  40 
lbs.     (5)  Each  =  39.16  lbs.     (6)  579.60  lbs.    (7)  15.59  lbs.  and  9  lbs. 

Pages  151,  152.     XXI.  Besolution  of  Forces  along  two  rectangular 
axes.    Articles  140,  141. 

(1)  R=  86.6  lbs,,  and  its  direction  makes  an  angle  of  150°  with 
P.  (2)  B  =  100  lbs.,  at  right  angles  to  P.  (3)  B  =  208.3 lbs.,  and 
makes  an  angle  of  298°  41'  (or  -  61°  19')  with  P.  (4)  i2  =  73.2 
lbs.,  and  acts  due  north.  (5)  B  =  346.4  lbs.,  and  acts  S.  54°  44'  E. 
(6),  the  forces  are  in  equilibrium.  (7)  B  =  61.22  lbs.,  and  acts 
S.  77°  30'  W. ;  the  system  will  be  kept  in  equilibrium  by  a  force  of 
61.22  lbs  acting  N.  77°  30'  R 


284  ANSWERS  TO  EXAMPLES. 

Pages  159,  160.     XXII.  Parallel  Forces.     Articles  143-149. 

(la)  n  =  12,  AC  =  28,  BC  =  20;  (lb)  R  =  30,  AB  =  135, 
BC  =  54;  (Ic)  R  =  6,  BC  =  40,  AG  =  IQ;  (Id)  M  =  8,  AB  =  48, 
BG  =  84.  (2cj)  Q  =  5,  ^(7  =  25,  BG=  15;  (26)  §  =  9,  AG=  27, 
^5  =  42;  (2c)  g  =  —  4,  5(7  =  40,  ^C  =  16;  {2d)  Q  =  -  4; 
AB  =  24,  BG  =  72.  (3)  At  A  36  lbs.,  at  B  12  lbs.  (4)  W  =  56 
lbs.,  at  C  24  lbs.  (5)  At  5  9  lbs.,  at  i>  1  lb.  (the  weight  of  the 
rod  is  neglected).  (6)  A  carries  84  lbs. ,  B  60  lbs.  (7)  At  J.  4  lbs. , 
at  B  and  C  8  lbs.  (8)  At  A  12  lbs.,  at  B  and  C  S  lbs.  (9)  Placed 
at  one  end.     (10)  31i  and  76^  lbs. 

Page  167.     XXIII.     Moments.    Articles  151-156. 

(l)72ft.lbs.  (2)  329.10  ft.lbs.  (3)  615.64  ft. lbs.  (4ci)  72  ft.lbs. 
m  55.15  ft.lbs. ;  (4c)  36  ft.lbs. 

Pages  180-182.    XXIV.  Centre  of  Gravity.    Articles  159-171. 

(1)  13i  in.  from  A.  (2)  6  in.  from  G  (3)  15  in.  from  D.  (4) 
1  in.  from  B.  (5)  2  in.  from  B.  (6)  2^  lbs.  (7)  4  oz.  (8)  i  lb. 
(9a)  A  carries  24  lbs.,  B  36  lbs. ;  {%)  A  should  stand  8  feet  from 
his  end.  (10)  1^  in.  from  the  centre  toward  A.  (11)  6  in.  from 
C,  on  a  line  making  an  angle  of  53°  8'  with  BG.  (12)  16  in.  from 
A,  on  a  line  bisecting  the  angle  BAG.  (13)  On  a  line  bisecting 
the  right  angle,  3  V2  in.  from  B.  (14«)  3  V2m.  from  the  centre 
toward  A\  (146)  4 i  in.  from  the  centre  on  a  line  drawn  to  the 
middle  point  of  AB;  (14c)  2  V2"from  the  centre  toward  B.  (15a) 
2|  in.  from  th^  centre,  on  the  line  drawn  to  the  middle  point  of 
CD;  (156)  2  V2'in.  from  the  centre  toward  C;  (15c)  f  V2"from  the 
centre  toward  C.  (16)  2f  in.  from  the  middle  point  of  BG,  on  the 
line  joining  the  centres  of  the  parallel  sides.  (17)  1  in.  from  the 
centre  of  the  original  circle.  (18a)  2|  in.  from  the  centre  of  the 
larger  circle.  (19)  ^  in.  from  the  centre  of  the  base.  (20)  8  in. 
from  B,  on  the  line  BD.  (21)  Hi  in.  from  the  vertex  of  the  tri- 
angle. (22)  On  the^axis,  4  in.  from  the  centre  of  the  larger 
cylinder. 

Pages  189, 190.    XXV.    StaUlity.    Articles  172-177. 

(la)  45  lbs.;  (16)  90  lbs.;  (Ic)  125  lbs.  and  250  lbs.     (2a)  42.43 
lbs.;  (26)  43.92  lbs.    (3)48°  11'.    (4)  80  feet  vertically.    (5a)  45"; 


ANSWERS  TO  EXAMPLES.  285 

(5b)  67°  10'  (the  vertex  lying  up  the  plane).  (6a)  16  lbs.;  (66)  48 
lbs.  (7a)  P=8  lbs. ;  (76)  20  lbs. ;  (7c)  61  lbs.  (8a)  15,406.6  ft.lbs. ; 
(8b)  49,442.7  ft.lbs. ;  (8c)  8781.8  ft.lbs. 

Pages  213, 214.    XXVI.    Lever,    Articles  185-190. 

(1)  160  lbs.  (2)  20  lbs.  (3)  12i  lbs.  (4)  200  lbs.,  100  lbs.,  87i 
lbs.  respectively.  (5)  87.12  lbs.  (6)  46.58  lbs.  (7)  180  lbs.  (8) 
67.12  vertical  (5),  and  163.71  lbs.  in  a  direction  making  an  angle  of 
12°  13'  with  a  vertical  line  through  F(7).  (9)  29.15  lbs.  (10)  38.18 
lbs.  (11)  96  and  48  lbs.  (12)  i  of  the  length  from  the  centre 
toward  the  end  having  the  heavier  weight.  (13)  8.24  feet  from 
the  end  at  which  the  force  8  acts. 

Page  214.     XXVII.    Balance.    Articles  191-193. 
(1)  14.07  lbs.    (2)  .837  :  1.    (3)  14.06  oz. 

Page  215.    XXVIII.    Steelyard.    Articles  194-196. 

(la)  The  zero  is  |  inch  from  C(  =  CD,  Fig.  139);  (lb)  f  in. 
from  G(-  GO,  Fig.  139);  (Ic)  the  graduation  is  to  12ths  of  an 
inch;  (Id)  1  lb.  and  20  lbs.  (2a)  i  in.  from  C(=  CD,  Fig.  138); 
(26)  li  in.  for  1  lb.;  (2c)  15^  lbs.  (3)  i  in.  and  ^V  in-  (4)  f  in. 
from  the  fulcrum.  (5)  6|  from  the  end  on  which  hangs  the 
weight.    (Qa)  18  in.  from  the  end  A  (Fig.  141);  (66)  14f  in.  from  J?. 

Page  220.     XXIX.    W heel  and  Axle.    Articles  202-208. 

(1)  W  ~  1200  lbs.  (2)  21,600  lbs.  (3)  1920  lbs.  (4)  P  =  88.54 
lbs.     (5)  8  feet  4  in.     (6)  P  =  10.61. 

Pages  234,  235.      XXX.  Pulley.    Articles  217-226 . 

(la)  50  lbs.   (16)  150  lbs.  (the  weight  of  the  platform  is  neglected 

in  each  case).     (2)  ti  =  4.     (3)  ?i  =  4.     (4)  ti  =  4.     (5)17=  193.18 

(30°),  =  173.2  (60°),  =  100  (120°).  =  51.76  (150°),  =  0  (180°).     (6) 

P=z  31.25,  and  31.72  lbs.  (7)  P=  125  and  125.25.  (8)  P=  33. 33  and 

32.97.  (9)  If  w'  is  the  weight  of  the  pulley  A,  and  w"  that  of  B,  then 

4  P-f-  w"  =  W-\-  w'.   (10)  Let  the  weights  of  the  movable  pulleys 

W        w' 
D,  C,  B,  A  be  respectively  lo',  lo",  w'",  w''",  then  P  —  -—-  -f-  -^7-  + 


286 


ANSWEES  TO  EXAMPLES. 


+ 


27 
P  = 


+  -^.   If  the  weights  of  the  pulleys  {w)  are  equal,  then 


1-  -gr) ;  or,  in  general,  i'  =  g^^ 


Pages  241-243.      XXXI.  IncUfied  Plane.    Articles  227-232. 

(la)  P=  41.04;  (lb)  P=  43.68;  (Ic)  P=  47.39.  {2a)  R  =  112.76; 
(25)  i2  =  127.7;  (2c)  R  =  89.07.  (3a)  575.9;  (3^>)  200;  (3c)  141.4; 
(dd)  115.5;  (3^)  101.5;  (3/)100.  (4a)  17.36;  (45)  50;  (4c)  70.7;  (4d) 
86.6;  {4e)  98.5;  (4/)100.  (5a)  567.1;  (55)  173.2;  (5c)  100;  (5d)  57.7; 
(5d)  17.6;  (5/)  0.  (6a)  17.6;  (65)  57.7;  (6c)  100;  (6(^)173.2;  (6^)567.1; 
(6/)  00.  (7)  T^^  of  a  ton.  (8)  35  and  21  lbs.  (9)11,464.  lbs.  (10) 
^  of  the  weight.  (11)  a  =  36°  52',  W  =  66|-.  (12)  5i.  (13) 
12  feet  and  18  feet. 


Page  233.     XXXII. 


Wedge.    Articles  233-235. 

(4)  They  are  equal.    (5)  133i 


(1)  115.2.    (2)  70.7.    j3)  38°  56' 
and  166|.    (6)  1  :  2  :  V3. 


Page  251.     XXXIII.     Screw.    Articles  236-242. 

(1)  9047.8  lbs.      (2)  .785  in.     (3)  12i  lbs.     (4)  7200  lbs.      (5) 
1  :  113.4.     (6)  2.    (7)  |  in.    (8)  81,430.3  lbs. 

Page  261.     XXXIV.  Pendulum.    Articles  24^247. 

(la)  9.78  in.;  (15)  13.03  feet;  (Ic)  20.37  feet.     (2)  39.21  in.     (3) 
13.01  feet.     (4)  About  91.2  feet.    (5)  38.3  times.    (6)  37.5  in. 


ANSWERS  TO  ADDITIONAL  EXAMPLES 

(ON  PAGES  263-278) 
INTRODUCING   THE  METRIC  UNITS. 


(1)  864  kilometers.  (2)  13f.  {da)  16f ;  (3&)  100.  (4m)  5.44  kilo- 
meters; (4J)  8|f  minutes.  (5)  2.56  kilometers.  (6a)  88  (per 
second);  (65)  3168  kilometers. 

IL,  A. 

(1)  The  distances  are  4.9,  19.6,  44.1,  78.4  meters;  the  velocities 
are  9.8,  19.6,  29.4,  39.2  meters  per  second.  (2a)  98  meters  per 
second;  (2b)  490  meters;  (2c)  93.1  meters.  (Sa)  4f  seconds.  (Sb) 
42  meters  per  second.  (4a)  7i  minutes;  (45)  275.6  meters,  (5a)  5 
seconds;  (5b)  122.5  seconds. 

II.,    B. 

(1)  8  meters-per-second  per  second.  (2a)  20  meters-per-second 
per  second;  (2b)  .36  kilometers;  (2c)  120  meters  per  second.  (3a) 
60  meters  per  second ;  (db)  150  meters.  (4)  Yes.  (5)  260  meters. 
(6)  20  meters-per-second  per  second. 

III. 

(la)  2  kilometers  per  hour;  (lb)  18;  (Ic,  Id)  8  and  24.  (2a) 
38°  40'  down  stream;  (2b)  128.1  meters  per  minute;  (2c)  20  minutes. 
(3a)  53°  8'  up  stream;  (36)  60  meters  per  minute;  (3c)  33J-  minutes. 
(4a)  13  meters  per  second;  (45)  N.  22°  37'  E.  (5a)  N.  67°  30'  E.; 
(55)  6.12  meters  per  second.  (6a)  5  meters  per  second;  (65)  36°  52' 
■with  his  direction. 


288       ANSWEES   TO   ADDITIONAL   EXAMPLES. 

IV. 

(1)  3  meter§  per  second.    (2)  4|  and  3|  meters  per  second.     (3) 
4.33  and  2.5.    (4)  12.99  and  7.5. 


(la)  4.9  meters-per-second  per  second;  (lb)  39.2;  (Ic)  19.6.  (2a) 
8  seconds;  (2b)  19.6  meters  per  second.  (3)  11.89  seconds.  (4a) 
1.4  meters-per-second  per  second;  (46)  90  meters. 

VI. 

(la)  80.6  meters  per  second;  (lb)  324.1  meters.  (2a)  5  seconds; 
(2b)  68.1  meters  per  second.  (3)  18.63  meters  per  second.  (4a) 
19.6  meters;  (4J)  3f  seconds.  * 

VII. 

(la)  and  (15)  5  seconds;  (Ic)  122.5  meters;  (Id)  44.1  and  4.9 
meters.  (2)  1  or  7f  seconds.  (3)  56  meters  per  second.  (4)  7^ 
seconds.    (5a)  28  meters  per  second;  (5b)  after  If  seconds  longer. 

VIII. 

(la)  12  seconds  ;  (16)  49.7  meters  per  second.    (2a)  29.6  meters 

per  second  ;  (2b)  79.2  meters.     (3a)  56  meters  per  second  ;  (36)  11^ 

seconds. 

IX. 

(la)  f  =  4  meters-per-second  per  second ;  (16,  Ic)  after  sliding  10 
seconds,  and  200  meters.  (2a)  20  seconds  ;  (26)  120  meters;  (2c) 
10.2  meters  per  second.     (3a)  28.8  kil.  per  hour  ;  (36)  480  meters. 

X. 

(la)  t  =  25  seconds;  (16)  5.304  kilometers.  (2)  24"  18'  or  65°  42'. 
(3)  u  —  280  meters  per  second.  (4)  After  3  seconds,  at  a  distance 
of  1.2  kilometers.     (5a)  200  meters  per  second;  (56)  122.5  meters. 

XI. 

(1)  2f  :  1.    (2)  2.73  :  1.     (3)  773.2.     (4)  13.394  kilograms. 

XII. 
fl)  4  times  the  earth's  radius.     (2)  277.3  meters-per-second  per 


ANSWERS  TO  ADDITIONAL  EXAMPLES.       289 

second.     (3)  About  1.6   meters-per-second  per  second.     (4)  2.7 
millimeters-per-second  per  second. 

XIIL 

(1)  5  meters  per  second.  (3)  3  meters  per  second.  (3)  10  meters 
per  second.  (4)  4,  3,  and  2.4  meters  per  second.  (5)  420.84  meters 
per  second. 

XIV. 

(la)  3.8  meters-per-second  per  second ;  (lb)  7.6  meters.  {2a)  At 
a  height  equal  to  If  the  radius  of  the  earth;  (2&)  2. 04  kilos.  (3) 
/  =  2.8;  tension,  4.286  kilos  (4)  20.4  seconds.  (5)  28|  kilos. 
{6a)  400  grams;  (6&)  392,000  dynes.  (7)  1.4  grams.  (8)  1.02  milli- 
grams. 

XV. 

(1)  25  kilos;  /  =  24.5.     (2)  8297|  kilos. 

XVI. 

(1)  ;M  =  i.  (2)  4.8  kilos.  (3)  160  grams.  {4a)  m  =  .36,  F=  5.07 
kilos.    (45)  fi  =  .36,  F=  10.14.    (5)  6.235  kilos. 

XVII. 

(1)  1  kilogram-meter  =  7.23  ft.lbs.  (2)  62,514  kilogram-meters. 
(3)  675,000  kilogram-meters.  (4)  100,000  kilogram-meters.  (5) 
172,000  kilogram-meters. 

XVIII. 

(1)  About  424  kilogram-meters.  (2)  200  kilogram-meters.  (3) 
50,000,000  kilogram-meters.  (4)  400,000  kilogram-meters.  (5)  270 
kilogram-meters.  (6)  4  kilometers  and  57^^  seconds.  (7)  50  kilos. 
(8)  150,000  kilos. 

XIX. 

(1)  i?  =  250  grams;  {a)  73°  44'.  (2)  11.14.  (3)  B  =  20.78,  r  = 
120°.    (4)  6.13  kilos.    (5)  910  grams.     (6)  8  kilos. 

XX.,  XXI. 

(1)  56.4  N.  20.52  E.  (2)  6.93  and  13.86.  (3)  25.21  kilos.  (4) 
295.44  kilos.    (5)  554.24  and  320  grams. 


290      ANSWEES  TO  ADDITIONAL  EXAMPLES. 

(1)  R  =  303.5,  the  angle  between  P  and  i?  is  -  10°  46'.  (2)  100 
kilos  in  the  same  direction  as  P. 

XXII. 

(1)  16  kilos  at  A,  and  8  at  B.  (2)  10^  at  C ;  W=  24^.  (3) 
A  carries  48  and  B  32  kilos.  (4)  On  A  2  kilos,  and  on  B  and  G 
each  4  kilos.    (5)  2  centimeters  from  one  end. 

XXIII. 

(1)  72  kilogram-meters.  (2)  30  kilogram-meters:  the  same  as 
before.  (3)  346.4  kilogram-meters.  (4)  25,  23.5,  and  8.55  kilo- 
gram-meters. 

XXIV.,  XXV. 

(1)  21  centimeters  from  A.  (2)  25  millimeters  from  B.  (3)  270 
grams.  (4)  10.02  gr.  (5)  125  from  the  centre  toward  A.  (6)  69  mil- 
limeters from  A,  on  the  line  drawn  to  the  middle  point  of  BC. 
(7a)  21  kilos;  {7b)  56  kilos.  (8a)  7.54  kilos  ;  (Sb)  38.63  kilos  ;  (8c) 
6.63  kHos.    (9a)  811.4  kilogram-meters  ;  {9b)  3090.2  ;  (9c)  450.85. 

XXVI. 

(1)  53i  kilos.  (2)  20  kilos.  (3)  3  kilos.  (4)  24.24  kilos.  (5) 
69.28  kilos.  (6)  15.81  kilos.  (7)  21.21  kilos.  (8)  1  meter  856.4  mm. 
from  the  force  8. 

XXVII.,  XXVIIL 

(1)  6  kilos  245  grams.  (2)  1  :  1.08.  (3)  1041.7  milligrams.  (4«) 
12  millimeters  from  C;  (4Z>)  80  millimeters;  (4c)  7.1  kilos.  (5)  25 
millimeters  from  the  fulcrum.  (6)  90  millimeters  from  the  end 
on  which  hangs  the  weight. 

XXIX,   XXX. 

(1)  900  kilos.  (2)  333i  kilos.  (3)  50.4  kilos.  (4)  2i  meters. 
-(5)  4  pulleys.  (6)  6  pulleys.  (7)  5  pulleys.  (8)  54.12  (2  a  = 
45^),  130.66  (2  a  =  135°).    (9)  50  kilos,  50.25  kilos. 

XXXI. 

(la)  60  kilos;  (1&)  69.28  kilos;  (Ic)  120  kilos.     (2a)  103.92;  (2^) 


ANSWERS  TO  ADDITIONAL   EXAMPLES.       291 

138.56;  (2c)  0.    (3a)  175.4;  (35)  93.34.    (4a)  20.52 ;  (45)  38.57.    (5)50 
kilos.    (6)  96,000  kilos. 

XXXII.,   XXXIIL 

(1)  38.64.  (2)  Each  100  kilos.  (3)  19°  12'.  (4)  100,  133*,  166f.— 
(5)  25,132.7  kilos.  (6)  31.42  millimeters.  (7)  15  kilos.  (8)  8333* 
kilos. 

XXXIV. 

(la)  .110  meter;  (lb)  8.939  meters;  (Ic)  2.235  meters.  (2)  .9962 
meters.    (3)  3.965  meters.    (4)48.8. 


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